Nicolas Franceschini
email: nicolas.franceschini@univmed.fr

Franck Ruffier

Julien Serres

Aerial navigation and optic flow sensing: a biorobotic approach

This chapter addresses some of the control problems involved in animal's visually guided 3D piloting. We formulated explicit control schemes which explain how insects may navigate without requiring any distance or speed measurements. The concept of the optic flow regulator, a feedback control system based on optic flow sensors, is presented. Based on a number of behavioral experiments conducted in our laboratory and elsewhere, we explain how OF regulators suffice to account for various insect flight patterns observed over the ground and over still water, under calm and windy conditions, and in straight or tapered corridors. These control schemes were tested in simulation and implemented onboard two types of insect-like robots, a micro-helicopter and a micro-hovercraft, which behaved very much like insects when placed in similar environments. These robots were all equipped with electro-optic OF sensors inspired by the results of our previous microelectrode studies on the motion sensitive neurons present in the flies' eyes. The simple and parsimonious control schemes described here do without any conventional avionic devices such as radio-altimeters, laser rangefinders, variometers, radars, sonars or GPS receivers. While these control schemes are little demanding in terms of neural resources -consistent with their integration in the lessthan-one-milligram insect's brain -they show great potential for simplifying the design of aerial and space vehicles, with interesting prospects in weight-reduction and low consumption.

Introduction

Insects are no more that kind of dumb creatures they used to be. Some insects have well developed learning and memory capacities whose essential mechanisms do not differ drastically from those of vertebrates [START_REF] Giurfa | The amazing mini-brain : lessons from a honeybee[END_REF]Menzel, 1997, Giurfa, 2003). Flying insects have been in the business of sensory-motor integration for more than hundred millions years. These star pilots navigate swiftly through most unpredictable environments, often attaining a level of agility that greatly outperforms that of both vertebrate animals and present day aerial robots. Insects are capable of dynamic stabilization, 3D autonomous navigation, ground avoidance, collision avoidance with stationary and nonstationary obstacles, tracking, docking, decking on movable substrates, autonomous takeoff, hovering, landing, etc. They also behave in a predictive manner, making the appropriate anticipatory postural adjustements that will allow them to take off in the right direction when they notice an approaching threat [START_REF] Card | Visually Mediated Motor Planning in the Escape Response of Drosophila[END_REF]. Insects' neural circuits are highly complex -commensurate with the sophisticated behavior they mediate -but unlike most (no less complex) vertebrate neural circuits, they can be investigated at the level of single, uniquely identifiable neurons, i.e., neurons that can be reliably identified in all the individuals of the species on the basis of their location in the ganglion, their exact shape and their consistent electrical responses. This great advantage of insect versus vertebrate neuroscience enables insect neuroscientists to accumulate knowledge during anything from a few days to several decades about a given individual neuron or a well defined neural circuit (Strausfeld, 1976[START_REF] Hausen | Neural mechanisms of course control in insects[END_REF][START_REF] Frye | Fly flight: A model for the neural control of complex behavior[END_REF]. This chapter deals with the autopilot systems that may allow insects to fly safely without bumping into things. The chapter summarizes our recent attempts to formulate explicit control schemes explaining how insects may navigate without requiring any distance or speed measurements. The aim of these studies was not to produce a detailed neural circuit but rather to obtain a picture that abstracts some basic control laws. We attempted to determine the variables the insect needs to measure, the variables it needs to control, and the causal and dynamic relationships between the sensory and motor variables involved. The last decades have provided evidence that flying insects guide themselves visually by processing the optic flow (OF) that is generated on their eyes as a consequence of their locomotion. In the animal's reference frame, the translational OF is the angular speed ω at which contrasting objects in the environment move past the animal as a consequence of locomotion [START_REF] Kennedy | Visual responses of flying mosquitoes[END_REF][START_REF] Gibson | The perception of the visual world[END_REF][START_REF] Lee | The optic flow field: the foundation of vision[END_REF][START_REF] Koenderink | Optic flow Vision Research[END_REF]. Our progress was achieved by performing simulation experiments and testing our control schemes onboard miniature aerial robots. Constructing a bio-inspired robot first requires exactly formulating the signal processing principles at work in the animal. It gives us, in return, a unique opportunity of checking the soundness and robustness of those principles by bringing them face to face with the real physical world (see also [START_REF] Webb | Can robots make good models of biological behavior ?[END_REF]. Like the early terrestrial Robot-Fly (the "robot-mouche": [START_REF] Pichon | Visual guidance of a mobile robot equipped with a network of self-motion sensors[END_REF][START_REF] Franceschini | Sequence-discriminating neural network in the eye of the fly[END_REF], our aerial robots are based on the use of electronic OF sensors [START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF][START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF] inspired by the housefly Elementary Motion Detectors (EMDs), which we previously studied in our laboratory (Rev. [START_REF] Franceschini | Early processing of colour and motion in a mosaic visual system[END_REF][START_REF] Franceschini | Sequence-discriminating neural network in the eye of the fly[END_REF][START_REF] Franceschini | Directionally selective motion detection by insect neurons[END_REF]. After recalling some aspects of the fly visual system and its motion sensitive neurons, we focus on the realisation of fly-inspired OF sensors. We then examine the problem of ground avoidance by flying creatures and introduce the principle of OF regulation. The microhelicopter demonstrator we built is then described, equipped with an electronic OF sensor and an OF regulator. The extent to which the OF regulator concept accounts for the actual behavioral patterns observed in insects is discussed. The results of recent behavioral experiments on honeybees trained to enter a corridor led us to introduce the dual OF regulator, an OF-based autopilot that is able to control both speed and clearance from the walls. This autopilot's performance was tested in simulation on a micro-hovercraft. We conclude by discussing the potential applications of these insect-derived principles to the navigation of aerial vehicles -in particular Micro Aerial Vehicles (MAVs) and Micro Space Vehicles (MSVs) which, like insects, are not supposed to carry the prohibitively large sensors and control systems present on conventional aircraft.

Figure 1: Head of the blowfly Calliphora erythrocephala with its panoramic compound eyes (left). The right part shows about 2% of the housefly retinal mosaic. A cluster of micrometer-sized photoreceptors is located in the focal plane of each facet lens : 6 outer receptors R1-6 surround a central cell R7 (prolonged by an R8, not seen here). These are the natural autofluorescence colors of the receptors observed in vivo under blue excitation (Franceschini et al., 1981a,b), after "optical neutralization of the cornea" [START_REF] Franceschini | Sa majesté des mouches[END_REF].

The fly visual system and its motion sensitive neurons

Each compound eye consists of an array of ommatidia, the frontend of which is a facet lens that focusses light on a small group of photoreceptor cells (Fig. 1, right). The fly retina is among the most complex and best organized retinal mosaic in the animal kingdom. It has been described in great details, with its different spectral types of photoreceptor cells, polarization sensitive cells and sexually dimorphic cells. There exists a typical division of labour within the retina (Rev. [START_REF] Franceschini | Chromatic organisation and sexual dimorphism of the fly retinal mosaic[END_REF][START_REF] Franceschini | Early processing of colour and motion in a mosaic visual system[END_REF][START_REF] Hardie | Functional organization of the fly retina[END_REF]:

• The two central photoreceptor cells, R7-8, display various spectral sensitivities that are randomly scattered across the retinal mosaic, as attested by the characteristic R7 autofluorescence colors (Fig. 1). R7 and R8 are thought to participate in color vision.

• The outer 6 photoreceptor cells (R1-R6) all have the same, dual-peak spectral sensitivity (blue-green + UV). They participate, in particular, in motion detection (Rev.: Buchner, 1984, Heisenberg and[START_REF] Heisenberg | Vision in Drosophila[END_REF][START_REF] Riehle | Motion detection in flies: parametric control over ON-OFF pathways[END_REF]. In this visual pathway, signal-to-noise ratio is improved by the presence of an ultraviolet sensitizing pigment that enhances their quantum catch [START_REF] Kirschfeld | Evidence for a sensitizing pigment in fly photoreceptors[END_REF], and by an exquisite optoneural projection called neural superposition [START_REF] Braitenberg | Patterns of projection in the visual system of the fly, I/Retina-Lamina projections[END_REF][START_REF] Kirschfeld | Die Projektion der optischen Umwelt auf das Raster der Rhabdomere im Komplexauge von Musca[END_REF][START_REF] Kirschfeld | Optische Eigenschaften der Ommatidien im Komplexauge von Musca[END_REF]. The R1-R6 photoreceptors therefore make for a high sensitivity ("scotopic") system [START_REF] Kirschfeld | Optische Eigenschaften der Ommatidien im Komplexauge von Musca[END_REF]). To estimate the OF, insects use motion sensitive neurons. Flies analyze the OF locally, pixelby-pixel, via a neural circuit called an "Elementary Motion Detector" (EMD). Further down the neural pathways, more precisely in that part of the third optic ganglion called the lobula plate (LP), about 60 large-field collator neurons called "lobula plate tangential cells" (LPTCs) integrate the outputs of large numbers of EMDs (Rev.: [START_REF] Hausen | Neural mechanisms of course control in insects[END_REF][START_REF] Egelhaaf | Movement detection in Arthropods[END_REF][START_REF] Hausen | Decoding of retinal image flow in insects[END_REF]. The LPTC neurons analyze the OF field generated by the animal's locomotion (Rev.: [START_REF] Krapp | Dendritic structure and receptive-field organisation of optic flow processing interneurons in the fly[END_REF]Borst andHaag, 2002, Taylor and[START_REF] Taylor | Sensory systems and flight stability : what do insects measure and why ?[END_REF]. Some of them transmit electrical signals via the neck to thoracic interneurons directly or indirectly responsible for driving the wing-, leg-, or head-muscles. Other LPTCs (in particular, the H1 neuron, Fig. 2b) send relevant signals to the contralateral eye. Taking advantage of the micro-optical techniques we had developed earlier (Rev. [START_REF] Franceschini | Sampling of the visual environment by the compound eye of the fly: fundamentals and applications[END_REF], we were able to activate a single EMD in the eye of the living housefly by stimulating single identified photoreceptor cells within a single ommatidium while recording the response of an identified motion sensitive neuron (H1) in the LP [START_REF] Riehle | Motion detection in flies: parametric control over ON-OFF pathways[END_REF][START_REF] Franceschini | Early processing of colour and motion in a mosaic visual system[END_REF], 1992 ;[START_REF] Franceschini | Directionally selective motion detection by insect neurons[END_REF]. A microelectrode (c) records the unit response (nerve impulses) of the motion sensitive neuron H1 to this "apparent motion" [START_REF] Franceschini | Directionally selective motion detection by insect neurons[END_REF].

We applied pinpoint stimulation to two neighboring photoreceptors (diameter ≅ 1µm) within the selected ommatidium (Fig. 2a) by means of a high-precision instrument (a hybrid between a microscope and a telescope: figure 2d), where the main objective lens was a single ocular facet (diameter≅ 25µm, focal length ≅ 50µm). This laboratory-made optical instrument served to : (i) select the facet lens (Fig. 2a), (ii) select two out of the seven receptors visible in vivo in the back focal plane of the facet lenslet, and (iii) stimulate these two photoreceptors (R1 and R6) successively with 1µm-light spots. Sequential stimulation produced an "apparent motion" that would simulate a real motion within the small visual field of the selected ommatidium. The H1-neuron responded with a vigorous spike discharge only when the motion was mimicked in the preferred direction (see Fig. 2c, top trace). The null response observed for the reverse sequence (Fig. 2c, bottom trace) attests to the remarkable sequence-discriminating ability of an EMD, which is indeed directionally selective [START_REF] Franceschini | Sequence-discriminating neural network in the eye of the fly[END_REF]. From many experiments of this kind, in which various sequences of light steps and/or pulses were applied to selected receptor pairs, an EMD block diagram was obtained and the dynamics and nonlinearity of each block were characterized [START_REF] Franceschini | Early processing of colour and motion in a mosaic visual system[END_REF][START_REF] Franceschini | Sequence-discriminating neural network in the eye of the fly[END_REF][START_REF] Franceschini | Directionally selective motion detection by insect neurons[END_REF]. While not unveiling the EMD cellular details -still elusive in both vertebrates and invertebrates -our analysis at least allowed the EMD principle to be understood functionally, paving the way for some transcription into another, man-made technology, such as electronics.

Neuromimetic optic flow sensors

In the mid 1980's, we designed a neuromorphic optic flow sensor [START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF][START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF], the signal processing scheme of which was inspired by what we had learned from the fly EMD. The OF is an angular speed ω [rad.s -1 ] that corresponds to the inverse of the time Δt taken by a contrasting feature to travel between the visual axes of two adjacent photoreceptors, separated by an angle Δϕ. Our OF sensor processes this delay Δt so as to generate a response that grows monotonically with the inverse of Δt, and hence with the optic flow ω (Fig. 3a). Short delays Δt give higher voltage outputs and vice versa. [START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF][START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF]. (b) purely analogue version (weight 5 grams) built in 1989 for Robot-Fly, whose compound eye housed a ring of 114 EMDs of this type [START_REF] Pichon | Visual guidance of a mobile robot equipped with a network of self-motion sensors[END_REF][START_REF] Franceschini | From insect vision to robot vision[END_REF]) (c) hybrid (analogue + digital) version (size : 7mm x 7 mm, mass 0.2 grams) based on a microcontroller and built using Low Temperature Co-fired Ceramics technology (LTCC) [START_REF] Pudas | A miniature bio-inspired optic flow sensor based on low temperature co-fired ceramics (LTCC) technology[END_REF].

Our scheme is not a "correlator scheme" (cf [START_REF] Hassenstein | Systemtheoretische Analyse der Zeitreihenfolgen und Vorzeichenauswertung bei der Bewegungsperzeption des Rüsszelkäfers Chlorophanus[END_REF]Reichardt, 1956, Reichardt, 1969) and corresponds to the class of "feature-matching schemes" [START_REF] Ullman | Analysis of visual motion by biological and computer systems[END_REF], where a given feature (here a change in intensity that may represent an edge) is extracted and tracked in time. The photodiode signal of each channel is first bandpass filtered (Fig. 3a) -resembling the analog signal measured in the large monopolar neurons of the fly lamina [START_REF] Zettler | Neuronal processing in the first optic neuropile of the compound eye of the fly[END_REF][START_REF] Laughlin | The role of parallel channels in early visual processing by the arthropod compound eye[END_REF]. The next step consists of hysteresis thresholding and generation of a unit pulse. In the EMD version built in 1989 for robot Fly (Fig. 3b), the unit pulse from one channel was sampling a long-lived decaying exponential function generated by the other channel [START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF][START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF], via a nonlinear circuit called a minimum detector, to give an output ω measured that grows as a monotonic function of the angular velocity ω=Δϕ/Δt (Fig. 3). The thresholding operation makes the voltage output virtually independent of texture and contrast -unlike the Reichardt correlator -and the circuit responds as well to natural scenes [START_REF] Portelli | An Insect-Inspired Visual Autopilot for Corridor-Following[END_REF]. A very similar EMD principle has been conceived, independently, a decade later by C. Koch's group at CALTECH, where it became known as the "facilitate and sample" velocity sensor [START_REF] Kramer | An analog VLSI velocity sensor[END_REF]. Yet another variant of our original "time of travel" principle was proposed another decade later (Möckel and Liu, 2006). Since the original analog design, we have built various versions of OF sensors based on this very principle. In the EMD currently used onboard our aerial robotic demonstrators, the signals are processed using a mixed (analog + digital) approach (Ruffier et al., 2003). Such OF sensors can be small and lightweight (the smallest one weighs only 0.2 grams: Fig. 3c) and several of them were also recently integrated on a miniature FPGA [START_REF] Aubépart | FPGA implementation of elementary motion detectors for the visual guidance of micro-air vehicles[END_REF](Aubépart et al. , 2008;;[START_REF] Aubépart | Bio-inspired optic flow sensors based on FPGA: application to micro-air vehicles[END_REF].

The problem of ground avoidance

To control an aircraft, it has been deemed essential to measure state variables such as groundheight, groundspeed, descent speed, etc. The sensors developed for this purposeusually emissive sensors such as radio-altimeters, laser rangefinders, Doppler radars, GPS receivers, forward-looking infrared sensors, etc. are far too cumbersome for insects or even birds to carry and to power. The OF sensors evolved by natural flyers over the last few hundred million years are at odds with these avionic sensors. They are based on vision, which is a nonemissive sensory system, and more precisely on OF sensors. The ventral OF experienced in the vertical plane by flying creatures -including aircraft pilots -is the apparent angular velocity ω generated by a point directly below on the flight track [START_REF] Gibson | Parallax and perspective during aircraft landings[END_REF][START_REF] Whiteside | Blur zone[END_REF]. As shown in figure 4a, the ventral OF depends on both the groundspeed V x and the groundheight h and is equal to the ratio between these two variables:

ω = V x / h [rad.s -1 ]
(Eq.1) [START_REF] Ibbotson | Evidence for velocity-tuned motion-sensitive descending neurons in the honeybee[END_REF], is able to measure the ventral OF, i.e., the angular speed ω at which any contrasting feature moves under the flying agent [START_REF] Franceschini | A bio-inspired flying robot sheds light on insect piloting abilities[END_REF].

We know that flies and bees are able to react to the translational OF independently of the spatial texture and contrast [START_REF] David | Compensation for height in the control of groundspeed by Drosophila in a new 'barber's pole' wind tunnel[END_REF][START_REF] Kirchner | Freely moving honeybees use image motion to estimate distance[END_REF][START_REF] Srinivasan | Range perception through apparent image speed in freely flying honeybees[END_REF], 1993;[START_REF] Baird | Visual control of flight speed in honeybees[END_REF]. We also know that some of their visual neurons may be involved in this reaction because they respond monotonically to w with little dependence on texture and contrast [START_REF] Ibbotson | Evidence for velocity-tuned motion-sensitive descending neurons in the honeybee[END_REF][START_REF] Shoemaker | Velocity constancy and models for widefield visual motion detection in insects[END_REF][START_REF] Srausfeld | Contrast sensitivity of insect motion detectors to natural images[END_REF]. Neurons facing downwards can therefore act as ventral OF sensors, and thus assess the V x / h ratio (Fig. 4).

Based on laboratory experiments on mosquitoes and field experiments on locusts, Kennedy put forward an "optomotor theory" of insect flight, according to which flying insects maintain a "preferred retinal velocity" with respect to the ground below [START_REF] Kennedy | Visual responses of flying mosquitoes[END_REF][START_REF] Kennedy | The migration of the desert locust (Schistocerca gregaria Forsk.) I. The behaviour of swarms[END_REF]. In response to wind, for example, insects may adjust their groundspeed or groundheight to restore the apparent velocity of the ground features. Kennedy's hypothesis has been repeatedly confirmed during the last 30 years: both flies and bees were found to maintain a constant OF with respect to the ground while cruising or landing [START_REF] David | The relationship between body angle and flight speed in free-flying Drosophila[END_REF][START_REF] Preiss | Set point of retinal velocity of ground images in the control of swarming flight of desert locusts[END_REF][START_REF] Srinivasan | Honeybee navigation en route to the goal: visual flight control and odometry[END_REF][START_REF] Srinivasan | How honeybees make grazing landings on flat surface[END_REF][START_REF] Baird | Visual control of flight speed and height in the honeybee[END_REF].

The problem is how insects may achieve this feat, since maintaining a given OF is a kind of chicken-and-egg problem, as illustrated by equation 1: an insect may hold its ventral OF, ω constant by adjusting either its groundspeed (if it knows its groundheight) or its groundheight (if it knows its groundspeed). Besides, the insect could maintain an OF of 1rad/s (i.e., 57°/s), for instance, by flying at a speed of 1m/s at a height of 1 meter or by flying at a speed of 2m/s at a height of 2m: there is an infinitely large number of possible combinations of groundspeed and groundheight that will give rise to the same "preferred OF".

Drawing on the experience we had with OF-based visual navigation of a terrestrial robot [START_REF] Pichon | Visual guidance of a mobile robot equipped with a network of self-motion sensors[END_REF][START_REF] Franceschini | From insect vision to robot vision[END_REF], we attempted early to develop an explicit flight control scheme for aerial navigation in the vertical plane. Our first tentative step on these lines was not particularly successful, because we were cornered in the general notion that prevailed in those days that insect navigation relies on gauging range [START_REF] Kirchner | Freely moving honeybees use image motion to estimate distance[END_REF][START_REF] Srinivasan | Range perception through apparent image speed in freely flying honeybees[END_REF][START_REF] Franceschini | From insect vision to robot vision[END_REF][START_REF] Srinivasan | How insects infer range from visual motion[END_REF]. In the experimental simulations we performed in 1994, for example [START_REF] Mura | Visual control of altitude and speed in a flying agent[END_REF], we assumed that the insect (or the robot) would know its groundspeed V x (by whatever means), so that by measuring ω it would be able to gauge the distance h from the ground (Eq.1) and react accordingly to avoid it. Although this procedure is still used in robotics (see, e.g., [START_REF] Barber | Autonomous landing of miniature aerial vehicles[END_REF]Srinivasan, 2006;[START_REF] Garratt | Vision-based terrain following for an unmanned rotorcraft[END_REF], where groundspeed can be determined (e.g., via GPS), this makes the way insects operate all the more elusive.

In 1999, we established (in simulation) how a rotorcraft (or an insect) might be able follow a terrain (Fig. 5a) and land (Fig. 5b) on the sole basis of OF cues, without measuring its groundspeed and groundheight [START_REF] Netter | Neuromorphic optical flow sensing for nap-of-the-Earth flight[END_REF]. [START_REF] Netter | Neuromorphic optical flow sensing for nap-of-the-Earth flight[END_REF]. (c) actual flight path of the tethered FANIA rotorcraft, which repeatedly jumped over a 30° rising terrain [START_REF] Netter | A robotic aircraft that follows terrain using a neuromorphic eye[END_REF].

The landing manoeuver was achieved under permanent visual feedback from a 20-pixel forward looking eye (with 19 EMDs). The driving force causing the progressive loss in altitude was the decrease in the horizontal flight speed, which occurred when the rotorcraft (or the insect) decreased its speed to land -either volontarily or in the presence of a headwind. The landing trajectory obtained in this simulation (Fig. 5b) already resembles the final approach of bees landing on a smooth surface [START_REF] Srinivasan | Honeybee navigation en route to the goal: visual flight control and odometry[END_REF]. The principle was first validated onboard FANIA, a miniature tethered helicopter having a single (variable pitch) rotor, an accelerometer and a forward looking eye with 20 pixels (and, therefore, 19 EMDs) arranged in the frontal meridian [START_REF] Netter | A robotic aircraft that follows terrain using a neuromorphic eye[END_REF]. This 0.8-kg rotorcraft had three degrees of freedom (surge, heave and pitch). Mounted at the tip of a flight mill, the robot lifted itself by increasing its rotor collective pitch. Upon remotely inclining the servovane located in the propeller wake, the operator made the helicopter pitch forward by a few degrees so that it gained speed and, therefore, climbed to maintain a reference OF with respect to the terrain below. FANIA was able to jump over a rising terrain by increasing its collective pitch as a function of the fused signals from its 19 EMDs (Fig. 5c).

The optic flow regulation principle

In spite of this early success to explain how an insect could navigate on an OF basis, we considered that Kennedy's insightful "optomotor theory" was calling for a clear formalization that would bring to light:

• the flight variables really involved • the sensors really required • the dynamics of the various system components • the causal and dynamic links existing between the sensory output(s) and the variable(s) to be controlled • the point of application of the various disturbances that an insect will experience in flight and the variables it will have to control to compensate for these disturbances.

We came up with an autopilot called OCTAVE (OCTAVE stands for Optical altitude Control sysTem for Autonomous VEhicles) that is little demanding in terms of neural (or electronic) implementation and could be just as appropriate for insects as it would be for aircraft (Ruffier & Franceschini, 2003). A ventral OF sensor was integrated into a feedback loop that would drive the robot's lift, and thus the groundheight, so as to compensate for any deviations of the OF sensor's output from a given set point (Ruffier and Franceschini, 2003, 2004a,b, 2005;[START_REF] Franceschini | A bio-inspired flying robot sheds light on insect piloting abilities[END_REF]. As we will see, this simple autopilot (Fig. 6a) enabled a microhelicopter to perform challenging tasks such as take-off, terrain following, reacting suitably to wind, and landing. The OCTAVE autopilot can be said to be an OF regulator. The word 'regulator' is used here as in control theory to denote a feedback control system designed to maintain an output signal constantly equal to a given set point. The Watt flyball governor from the 18 th century, for instance, was not only one of the first servomechanisms ever built: it was also the very first angular speed regulator. It served to maintain the rotational speed of a steam engine shaft at a given set point, whatever interferences occurred as the result of unpredictable load disturbances. The Watt regulator was based on a rotational speed sensor (meshed to the output shaft), whereas the OF regulator is based on a noncontact rotational speed sensor -an OF sensor -that measures the ventral OF (again in rad/s). Specifically, the OF signal ωmeas delivered by the OF sensor (see Fig. 6a) is compared with the OF set point, ωset. The comparator produces an error signal: ε = ωmeasωset which drives a controller adjusting the lift L, and thus the groundheight h, so as to minimize ε. All the operator does is to set the pitch angle Θ and therefore the airspeed (see figure 6A): the OF regulator does the rest, that is, it attempts to keep ω constant by adjusting the groundheight h proportionally to the current groundspeed V x . In the steady state (t = ∞), ω meas ≅ ω set , and the groundheight becomes proportional to the groundspeed .

h = K V x (with K = 1/ω SET = constant) (Eq. 2)
The controller includes proportional and derivative (PD) functions, which ensure closed-loop stability in the groundspeed range of 0-3m.s -1 . Controlling F (via the rotor speed) actually affects not only L but also T. This coupling is negligible, however, because Θ is always ≤ 10° for the highest speed attained (3ms -1 ), so that the ensuing change in L is at least 6-times (cotan 10°) greater than the change in T.

A micro-helicopter equipped with an OF sensor and an OF regulator

We tested the idea that insects may be equipped with a similar OF regulator by comparing the behavior of insects with that of a "seeing helicopter" placed in similar situations. The robot we built (Fig. 7a) is a micro-helicopter (MH) equipped with a simple, 2-pixel ventral eye driving an EMD acting as an OF sensor (Fig. 7a). The 100-gram robot is tethered to an instrumented flight mill consisting of a light pantographic arm driven in terms of its elevation and azimuth by the MH's lift and forward thrust, respectively (Fig. 7b). The flight mill is equipped with ground-truth azimuthal and elevation sensors with which the position and speed of the MH can be monitored accurately in real time (Ruffier and Franceschini, 2003).

Any increase in the rotor speed causes the MH to rise, and the slightest (operator mediated) forward ("nose-down") tilting by a few degrees produces a forward thrust component that causes the MH to gain forward speed. The flight mill is equipped with ground-truth azimuthal and elevation sensors that allow the position and speed of the MH to be monitored at high accuracy and in real time. Since the MH purpose was to demonstrate a basic principle, it was equipped with an elementary ventral eye composed of only two photoreceptors driving a single Elementary Motion Detector (EMD) built according to the principle shown in Fig.3a (Ruffier andFranceschini, 2003, Ruffier et al., 2003).

Micro-helicopter's versus insects' behavioral patterns

OCTAVE's OF regulator scheme results in the behavioral patterns shown in Fig. 8. Between arrowheads 1 and 2, the operator simply pitched the MH forward rampwise by an angle ΔΘ = +10°. The ensuing increase in groundspeed V x (up to 3m.s -1 , see Fig. 8B) automatically made the MH take off, since the feedback loop consistently increased h proportionally to V x to comply with equation (3). [START_REF] Franceschini | A bio-inspired flying robot sheds light on insect piloting abilities[END_REF].

Once reaching a constant speed, the MH flew level at a groundheight of approximately 1mthe value imposed by the OF set point ω set = 3rad.s -1 = 172°.s -1 (Fig. 8C). After covering 42m, the MH was pitched backwards rampwise by an opposite angle ΔΘ = -10° (between arrowheads 3 and 4), and the ensuing deceleration (see Fig. 8B) effectively produced an automatic descent and landing -a safe landing maneuver since it is performed "under visual control". The actual OF (Fig. 8D) is seen to have been held relatively -but not perfectlyconstant throughout the journey, even during the takeoff and landing maneuvers where the groundspeed can be seen to vary considerably (Fig. 8B).

Figure 9 shows that the various robot's flight patterns are extremely robust and reproducible, including those over a rising terrain and in the presence of wind. The OF regulator concept was found to account for a series of puzzling, seemingly unconnected flying abilities observed in various insect species, as summarized below: (details in [START_REF] Franceschini | A bio-inspired flying robot sheds light on insect piloting abilities[END_REF]):

• automatic terrain following. A gradual increase in relief constitutes a "disturbance" that impinges on the system at a particular point (see Fig. 7a). The closed feedback loop overcomes this disturbance by increasing the flight altitude, resulting in a constant groundheight over the rising terrain (Fig. 9b). This may account for the well-documented terrain and canopy following abilities of migrating insects (e.g., [START_REF] Srygley | Orientation mechanisms and migration strategies within the flight boundary layer[END_REF].

• suitable reactions to headwind. is a disturbance that impinges on the system at a different point (see Fig. 6a) and reduces the groundspeed. The feedback loop overcomes this disturbance by forcing the robot to descend (and even to land smoothly by strong wind) (Fig. 9d). A similar reaction was observed in locusts, honeybees and dung beetles.

• flying over a no-contrast zone. Here the OF sensor fails to respond, which irremediably causes the robot to crash, just as honeybees crash into mirror-smooth water [START_REF] Heran | Windkompensation und Seitenwindkorrektur der Bienen Flug über Wasser[END_REF].

• landing on a flat surface. During the final approach, which starts when the MH has regained its completely upright position (arrowhead 4 in Fig. 8a and vertical line in Fig. 9c), the OF regulator forces the MH to land smoothly at a constant descent angle, α, which can be calculated as follows:

(Eq. 3)
where ω set [rad.s -1 ] is the OF set-point, and τ [s] is the MH's surge time constant (τ MH  = 2.15s). Interestingly, honeybees were also observed to land with a constant slope on flat surfaces, but the way they may achieve this feat has been explained quite differently. According to [START_REF] Srinivasan | How honeybees make grazing landings on flat surface[END_REF], landing bees would follow two rules: "(a) adjusting the speed of forward flight to hold constant the angular velocity of the image of the surface as seen by the eye, and (b) making the speed of descent proportional to the forward speed". By contrast, the OF regulator automatically genrates smooth landing with a constant slope while adjusting neither the forward speed nor the speed of descent. The OF set point ω set is the reference signal that the feedback loop must maintain constant under all circumstances. ω set would thus correspond to Kennedy's early ideas about a "preferred angular velocity of image movement" [START_REF] Kennedy | Visual responses of flying mosquitoes[END_REF][START_REF] Kennedy | The migration of the desert locust (Schistocerca gregaria Forsk.) I. The behaviour of swarms[END_REF]. By increasing its ω set (see Fig. 7A), the flier will follow the terrain at a lower height, as typically observed in the case of the MH (see Ruffier and Franceschini, 2003, Fig.9a). Conversely, by setting ω set near zero, the animal will rise farther from the ground, as also typically observed on the MH. This OF set point ω set may well be under the control of other (e.g., visual or olfactory) sensors, but it may also depend on other external and/or internal factors such a 'drive for migration' (in the case of locusts: [START_REF] Preiss | Set point of retinal velocity of ground images in the control of swarming flight of desert locusts[END_REF] Taken together, these data show that an aerial vehicle equipped with an OF regulator will be able to take off, navigate over flat or rising terrain, react sensibly to headwind and land without ever measuring and holding any groundheights, groundspeeds or descent speeds. A similar principle was recently applied to a 0.4kg flying wing, which by flying at a constant cruise speed (imposed by an extra airspeed regulator) was able to maintain its altitude from its ventral OF measurement [START_REF] Beyeler | Vision-based control of near-obstacle flight[END_REF].

! " = #arctan 1 $ set % & ' ( ) * +

Micro-hovercraft's versus honeybees' behavioural patterns

Behavioral experiments on several insect species have long shown that motion perceived by the lateral part of their compound eyes affects the forward thrust, and hence the forward speed (rev. : Franceschini et al., 2007). Firstly, when flying through a tapered corridor, honeybees slowed down as they approached the narrowest section and speeded up when the corridor widened beyond this point [START_REF] Srinivasan | Honeybee navigation en route to the goal: visual flight control and odometry[END_REF]. The authors concluded that bees tended to adjust their speed proportionally to the local corridor width by regulating the image velocity. Secondly, when flying through a straight corridor, honeybees tended to fly along the midline [START_REF] Srinivasan | Range perception through apparent image speed in freely flying honeybees[END_REF]. To explain this "centering behavior", the authors hypothesized that bees were balancing the speeds of the retinal images (i.e., the lateral OFs) of the two walls, an hypothesis that subsequently gave rise to many wheeled and aerial robots capable of centering in a corridor (e.g., [START_REF] Coombs | Centering behavior using peripheral vision[END_REF]Duchon and Warren, 1995;[START_REF] Santos-Victor | Divergent stereo in autonomous navigation: from bees to robots[END_REF]. We recently found, however, that honeybees trained to fly along a larger corridor do not systematically center on the corridor midline (Fig. 10b,c). They keep remarkably close to one wall, even when part of the opposite wall is missing (Fig. 10d). Distance from that wall (D R or D L ) and forward speed V x were on average such that the speed-to-distance ratio (i.e., the lateral OF) was maintained practically constant, at about 230°/s in our 95-cm wide corridor (Serres et al., 2008a). (Serres et al., 2008a).

With a view to explaining the various honeybees' behaviors observed in the various corridors, we came up with the design of the LORA III autopilot (LORA III stands for Lateral Optic flow Regulator Autopilot, mark III), which is able to control both the forward speed V x of an aerial vehicle and its lateral distances D R and D L from the two corridor walls jointly, without ever measuring speed and distances (Fig. 11A) (Serres et al., 2008b). A micro-hovercraft (HO) was used, which can produce insect-like forward and sideward slips independently, because the two added lateral thrusters LT1 and LT2 (Fig. 11B,C) make it fully actuated. It travels at a constant altitude (~2mm) and senses the environment with two laterally oriented eyes measuring the right and left OFs. The HO's heading is maintained along the X-axis of the corridor (Fig. 11C) by a heading lock system which compensates for any yaw disturbances by controlling the two rear thrusters differentially. This system mimics the honeybee's heading lock system, which is based on a polarized light compass (Rossel and Wehner, 1972) and gives the insect an impressively straight course even in the presence of wind [START_REF] Riley | Flight trajectories of foraging insects: observations using harmonic radar[END_REF] . (Serres et al., 2008b).

LORA III autopilot is based on only two OF sensors (one looking to the right, one to the left). The forward speed V x is controlled (via a proportional-integral controller) by the error signal ε Fwd  between the sum of the two OFs (right and left) and the forward OF set-point ω setFwd .

The side slip speed V y , and thus the lateral distances D L and D R to the walls, is controlled (via a proportional derivative controller) by the error signal ε Side  between the larger of the two OFs and the sideways OF set-point ωsetSide. The HO's behavior is entirely determined by these two OF set points: ω setFwd = 300°/s and ω setSide = 230°/s. Simulation experiments where the HO navigated in a straight or tapered corridor accounted for all the honeybees' behaviors reported above (Serres et al., 2008b). Fig. 12 illustrates, for example, the HO's behavior in a tapered corridor. Whatever the position of its initial ordinate y 0 , the HO automatically decelerated on approaching the narrowest section of the corridor and accelerated when the corridor widened. It can be seen that the forward control system succeeded in keeping the sum of the two lateral OFs measured nearly constant and equal to the forward OF set-point ω setFwd = 300°/s (Fig. 12c). Likewise, the side control system succeeded in keeping the larger of the two lateral OFs measured practically constant and equal to the sideways OF set-point ω setSide = 230°/s (Fig. 12e). The result is that the HO automatically tuned both its groundspeed an its distance to the walls jointly, without any knowledge about the current corridor width, the groundspeed and the clearance from the walls. The field of view (FOV) of the eyes and the provocatively small numbers of pixels (four) and EMDs (two) will obviously need to be increased for dealing with navigation in more sparsely textured environments. It will also be necessary to control the heading direction (e.g., [START_REF] Zufferey | Bio-inspired flying robots[END_REF] to enable the HO to successfully negotiate more challenging corridors including L-junctions and T-junctions (cf Humbert and Hyslop, 2010)

Conclusion

In studying the types of operations that insects may perform to guide their flight on the basis of optic flow (OF) cues, we came up with several bio-inspired autopilot principles that harness the power of the translatory OF parsimoniously and therefore offer interesting prospects for MAV autonomous guidance. The micro-helicopter's outstanding visuo-motor performance (Fig. 7,8) suggests how insects and MAVs may take off, follow terrain and land if they are equipped with OF sensors facing the ground and an OF regulator that servoes the measured OF to a given set point (Fig. 6). The great advantage of this autopilot is that it requires neither measuring nor computing nor regulating (i.e., holding constant) any groundspeeds or groundheights. The only variable it needs to measure and regulate is the OF -a variable that can be accessed straight forward by dedicated sensors called OF sensors (Fig. 3). The recent finding that bees gradually descend when they fly in a corridor the floor of which is moved in the same direction as their flight, is fully consistent with the OCTAVE autopilot model (Portelli et al., 2010a). The OCTAVE principle differs markedly from another OF-based navigation strategy which requires measuring not only the ventral OF but also the groundspeed (e.g., from GPS), with an aim to estimate the groundheight (see equation ( 2)) [START_REF] Barber | Autonomous landing of miniature aerial vehicles[END_REF]Srinivasan et al., 2006, Garratt and[START_REF] Garratt | Vision-based terrain following for an unmanned rotorcraft[END_REF]. OCTAVE and LORA III autopilots consist of interdependent OF regulators, which control the lift, lateral thrust, and forward thrust, on which the groundheight, lateral position and groundspeed, respectively, depend. The block diagrams (Fig. 6 and Fig. 10) show which variables need to be measured, which ones are controlled and which ones are regulated, as well as the point of application of the various disturbances. In contrast with the control schemes presented by [START_REF] Conroy | Implementation of wide-field integration of optic flow for autonomous quadrotor navigation[END_REF] for a quadrotor navigating a straight corridor, OCTAVE's task is not to hold a "desired height" and LORA III's task is not to hold a "desired forward speed" and a "desired lateral position" in the corridor. Before entering the corridor, our hovercraft does not know how fast it will eventually fly and whether or not it will center or follow one wall: it is the environment itself that constrains both its speed and its distance to the walls, depending on the OF set points (Fig. 12). These explicit control schemes account for a series of flying abilities observed in numerous insect species, including honeybees' habit of landing at a constant slope and their flight pattern along a straight or tapered corridor. Our recent finding that bees do not center systematically in a corridor but may follow a wall unilaterally (Fig. 9d) cannot be accounted for by the "optic flow balance hypothesis" [START_REF] Srinivasan | Range perception through apparent image speed in freely flying honeybees[END_REF] but is convincingly accounted for by the LORA III model, where "centering behavior" arises as a particular case of "wall-following behavior" (Serres et al., 2008b). OCTAVE and LORA III interdependent OF regulators were recently combined in a comprehensive model, called ALIS (AutopiLot using an Insect-based vision System), which allows a simulated bee to travel safely along (horizontally or vertically) tapered tunnels, by automatically controlling both its speed and its clearance from the right wall, left wall, ground and roof, without ever estimating any groundspeeds or distances (Portelli et al, 2010). The simulated bee navigates on the basis of two parameters alone: the forward OF set-point and the positioning OF set-point. While it is conceivable that on board insects the OF set points may depend on either innate, internal or external factors, MAVs could receive their OF set-points from the control station via a radiolink. Once engineered further, the OF-feedback control systems we have described here could serve to improve navigation aids and automatic maneuvers. When measured on a commercial aircraft, a ventral OF greater than the set point would mean that the current altitude is "too low for the current groundspeed" [START_REF] Franceschini | Steering aid system for altitude and horizontal speed, perpendicular to the vertical, of an aircraft, and aircraft equipped therewith[END_REF]. Installed on board a MAV, the OCTAVE autopilot would enable it to automatically take off, fly level at a groundheight proportional to the groundspeed, follow shallow terrains, make it react automatically to unpredictable headwind or tailwind disturbances (by descending or climbing, respectively), and land safely [START_REF] Franceschini | Steering aid system for altitude and horizontal speed, perpendicular to the vertical, of an aircraft, and aircraft equipped therewith[END_REF], Ruffier and Franceschini, 2004, 2005b). Both OCTAVE and LORA III could potentially guide MAVs indoors or through complex terrains such as mountain or urban canyons, without using any computationally intensive visuomotor algorithms. Since these autopilots do not rely on GPS and ILS, and do not require any bulky (and power-hungry) emissive sensors such as RADARs, SONARs, FLIRs, or laser rangefinders, they can be of very small size, lightweight and power-lean, and therefore meet the challenge of future insect-scale MAVs. For the same reasons, these simple autopilots could potentially be adapted to micro-or nano-space vehicles (MSVs) landing on command on other celestial bodies or performing rendezvous and docking missions in space. A recent paper presented the first simulation experiment conducted in our laboratory, showing an OCTAVE-based descent to the Moon [START_REF] Valette | Biomimetic optic flow sensing applied to a lunar landing scenario[END_REF]. Over the last 2.6 km, the lander actuated its thrusters as a function of its pitch, so as to brake from 1800 km/h down to only 15km/h. This descent (from 500m down to 10m above the Moon surface) occurred "under visual control", thanks to the ventral OF sensor and the OF regulator, without using any altimeters or speedometers. Throughout descent, the lunar lander maintained an approximately constant OF of about 0.3rad.s -1 (i.e., 17.2° s -1 ). An aerial Martian rover equipped with more elaborate OCTAVE and LORA III autopilots could take off autonomously and explore an area, skimming the ground and hugging the walls of a canyon, and adapting its groundspeed and clearance from the walls automatically to the width of the canyon. With ground and obstacle avoidance at a premium, the rover could perform its mission successfully in uncharted environments, despite being unaware of its own groundspeed and altitude at all times. The orbiter (or the Earth-based control station) would simply be required to send the rover a set of low bandwidth signals: the values of the OF set points [START_REF] Franceschini | Towards Automatic Visual Guidance of Aerospace Vehicles : From Insects to Robots[END_REF].

Figure 2 :

 2 Figure 2: (a-c) Principle of the experiment aimed at deciphering the principle of motion vision in flies, using optical stimulation of single photoreceptors. (d) Triple-beam incident light "microscope-telescope" that delivers a 1µm light spot to two neighboring photoreceptor cells, R1 and R6, successively (see (a)).A microelectrode (c) records the unit response (nerve impulses) of the motion sensitive neuron H1 to this "apparent motion"[START_REF] Franceschini | Directionally selective motion detection by insect neurons[END_REF].

Figure 3 :

 3 Figure 3: (a) Principle of the electro-optic flow sensor derived from our electrophysiological analyses of the housefly's EMD[START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF][START_REF] Franceschini | Passive, noncontact optical velocity sensor Technical Report ANVAR/DVAR[END_REF]. (b) purely analogue version (weight 5 grams) built in 1989 for Robot-Fly, whose compound eye housed a ring of 114 EMDs of this type[START_REF] Pichon | Visual guidance of a mobile robot equipped with a network of self-motion sensors[END_REF][START_REF] Franceschini | From insect vision to robot vision[END_REF]) (c) hybrid (analogue + digital) version (size : 7mm x 7 mm, mass 0.2 grams) based on a microcontroller and built using Low Temperature Co-fired Ceramics technology (LTCC)[START_REF] Pudas | A miniature bio-inspired optic flow sensor based on low temperature co-fired ceramics (LTCC) technology[END_REF].

Figure 4 :

 4 Figure 4: Definition (a) and measurement of the ventral optic flow ω experienced by an insect (or a robot) flying in translation in the vertical plane. (b) an EMD of the type shown in Fig.3, like the EMDs driving the honeybees' VT neurons[START_REF] Ibbotson | Evidence for velocity-tuned motion-sensitive descending neurons in the honeybee[END_REF], is able to measure the ventral OF, i.e., the angular speed ω at which any contrasting feature moves under the flying agent[START_REF] Franceschini | A bio-inspired flying robot sheds light on insect piloting abilities[END_REF].

Figure 5 :

 5 Figure 5: Optic flow based terrain following and landing of a miniature helicopter whose eye (20 pixels, 19 EMDs) covers a frontal FOV of 75° in the vertical plane, centered at 40° below the horizon (a) Simulation of nap-of-the-earth flight (initial conditions: height: 5m, speed 2m.s -1 ; iteration step 1s). (b) Landing initiated by simply decreasing the rotorcraft horizontal speed at every iteration while retaining the same request to maintain a reference OF value[START_REF] Netter | Neuromorphic optical flow sensing for nap-of-the-Earth flight[END_REF]. (c) actual flight path of the tethered FANIA rotorcraft, which repeatedly jumped over a 30° rising terrain[START_REF] Netter | A robotic aircraft that follows terrain using a neuromorphic eye[END_REF].

Figure 6 :

 6 Figure 6: (A)The OCTAVE optic flow regulator (bottom feedback loop) controls the mean flight force vector and hence the lift, and hence the groundheight, so as to maintain the ventral optic flow ω constant and equal to the set-point ω set . (B) Flies, like helicopters, pitch forward slightly to increase their forward thrust, and hence their airspeed. As long as they pitch forward by Θ <10° , the lift component L does not incur any major loss[START_REF] Franceschini | A bio-inspired flying robot sheds light on insect piloting abilities[END_REF].

Figure 7 :

 7 Figure 7: (a) 100-gram Micro-helicopter (MH) equipped with a ventral OF sensor (Fig.3e) and the OF regulator shown in A. (b)The MH can be remotely pitched forward by a small angle Θ while keeping its roll attitude. It lifts itself and circles counterclockwise at speeds up to 3m/s and heights up to 3m over a large arena (outside diameter: 4.5m), giving rise to the flight patterns given in Figs.8-9The flight mill is equipped with ground-truth azimuthal and elevation sensors with which the position and speed of the MH can be monitored accurately in real time(Ruffier and Franceschini, 2003).

Figure 8 :

 8 Figure 8: Flight variables monitored during a 70-meter flight of the micro-helicopter over a flat, randomly textured pattern (shown at the bottom of (A)), including take-off, level flight and landing at a constant descent angle. (a) Flight path consisting of about six laps on the circular test arena. (B) Groundspeed V x . (C) Output ω meas of the OF sensor. (D) Actual OF, ω (calculated as V x /h) resulting from the behavioral reaction[START_REF] Franceschini | A bio-inspired flying robot sheds light on insect piloting abilities[END_REF].

Figure 9 :

 9 Figure 9: Performance and reproducibility of the micro-helicopter's flight path in the longitudinal plane. (a) automatic take-off obtained by gradually pitching the rotorcraft forward (nose-down) by 10° from the vertical. (b) automatic terrain following over the rising terrain shown in Fig.6d. (c) automatic approach and landing upon gradually raising the rotor axis vertically (nose-up). (d) a light headwind made the rotorcraft descend (dotted trajectory). A strong headwind forced the MH to land (continuous curve) at negligible vertical and forward speeds(Ruffier and Franceschini, © 2004 IEEE)

Figure 10 :

 10 Figure 10: Honeybees' centering and wall-following behaviors. Bees were trained to enter a wide (width 0.95m) 3-meter long corridor, formed by two 0.25-m high walls lined with vertical white-and-grey stripes (period 0.1m; contrast m = 0.41). The bee's entrance (E C ) and the feeder (F C ) were placed either on the corridor midline (a) or on one side (b, c, d). In (d), part of the left wall was removed during the trials. Mean distances distribution as shown on top(Serres et al., 2008a).

Figure 11 :

 11 Figure 11: (A) LORA III autopilot enabling a hovercraft to navigate in a corridor by controlling its forward speed and its distance to the walls jointly, without measuring any speeds or distances. (B, C) fully actuated hovercraft (HO) equipped with two OF sensors looking sidewards at an angle of +90/-90°. The corridor walls are randomly textured(Serres et al., 2008b).

Figure 12 :

 12 Figure 12: Simulated navigation of the hovercraft (HO) along a tapered corridor, requiring no data on the corridor width, tapering angle α, nor any measurements of groundspeed and distance to the walls. (a) Simulated trajectories of the HO moving to the right along the corridor (tapering angle α = 7°) with three initial ordinates y 0 (open dots: y 0 =0.90m, crosses: y 0 =0.60m, full dots: y 0 = 0.30m). (b) Forward speed profile. (c,d) sum and maximum of the two lateral OFs measured, respectively. (e,f) sum and maximum of the two actual OFs, respectively (Serres et al. 2008b).

Acknowledgements

We are grateful to S. Viollet, F. Aubépart, G.P., Masson, D. Dray, L. Kerhuel and G. Portelli for their help and fruitful suggestions during this research. We are also thankful to M. Boyron and F. Paganucci for their expert technical assistance and J. Blanc for revising the English manuscript. The work presented here was supported by CNRS (Life Science and Engineering Science), EU contracts (IST/FET 1999-29043 and IST/FET 2009-237940), DGA contracts (05-34022 and 0451037) and an ESA contract (08-6303b).


Ning Dang

Shuai Chen

Lingyu Liu

Eric Maire
email: eric.maire@insa-lyon.fr

Jérôme Adrien

Sophie Cazottes

Wenlong Xiao
email: wlxiao@buaa.edu.cn

Chaoli Ma

Lian Zhou

Analysis of hybrid fracture in α/β titanium alloy with lamellar microstructure

Keywords: Crystal plasticity, Spectral method, Fracture behavior, Titanium alloys, Deformation heterogeneities

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Owing to their low density, high specific strength, and excellent corrosion resistance, two-phase α/β titanium alloys are widely used in modern industrial applications, such as airplane components, medical devices and implants [START_REF] Peters | Titanium and Titanium Alloys: Fundamentals and Applications[END_REF][START_REF] Boyer | Materials Properties Handbook: Titanium Alloys[END_REF][START_REF] Jia | A novel Ti-6Al-4V alloy microstructure with very high strength and good ductility[END_REF]. At ambient temperature, these alloys consist predominantly of an HCP α-phase and a BCC β-phase. The heterogeneous deformation behavior of the two phases leads to plastic incompatibilities [START_REF] Boyer | Materials Properties Handbook: Titanium Alloys[END_REF][START_REF] Jia | A novel Ti-6Al-4V alloy microstructure with very high strength and good ductility[END_REF] which promote a damage initiation in the form of void nucleation and growth. The α/β interface decohesion [START_REF] Faleskog | Micromechanics of coalescence-I. Synergistic effects of elasticity, plastic yielding and multi-size-scale voids[END_REF][START_REF] Helbert | The influence of stress trlaxiality on the damage mechanisms in an equiaxed α/β Ti-6AI-4V alloy[END_REF][START_REF] Tirry | A multi-scale characterization of deformation twins in Ti6Al4V sheet material deformed by simple shear[END_REF][START_REF] Pineau | Failure of metals I: Brittle and ductile fracture[END_REF] occurs especially in alloys with long straight interfaces(full lamellar microstructure) [START_REF] Dang | Analysis of shear stress promoting void evolution behavior in an α/β Ti alloy with fully lamellar microstructure[END_REF][START_REF] Dang | Three dimensional X-ray tomographic analysis of crack nucleation and growth by void coalescence in an α/β titanium alloy with lamellar microstructure[END_REF]. To expand the future use of these alloys, it is necessary to understand well the relation between plastic inhomogeneity and damage initiation. Furthermore, because of the strong dependence of the plastic behavior on the microstructure, the microstructure morphology should be taken into consideration. As demonstrated by many investigations [START_REF] Schoenfeld | Texture effects on shear response in Ti-6Al-4V plates[END_REF][START_REF] Thomas | Image-based crystal plasticity FE framework for microstructure dependent properties of Ti-6Al-4V alloys[END_REF][START_REF] Hasija | Deformation and creep modeling in polycrystalline Ti-6Al alloys[END_REF][START_REF] Khan | Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys[END_REF][START_REF] Deka | Crystal plasticity modeling of deformation and creep in polycrystalline Ti-6242[END_REF][START_REF] Venkatramani | A size-dependent crystal plasticity finite-element model for creep and load shedding in polycrystalline titanium alloys[END_REF][START_REF] Khan | Multiaxial and non-proportional loading responses, anisotropy and modeling of Ti-6Al-4V titanium alloy over wide ranges of strain rates and temperatures[END_REF][START_REF] Dunne | Lengthscale-dependent, elastically anisotropic, physically-based hcp crystal plasticity: Application to cold-dwell fatigue in Ti alloys[END_REF], realistic microstructure-based crystal plasticity(CP) methods are suitable to investigate the role of the microstructure regarding plasticity deformation and damage evolution in metallic materials [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications[END_REF][START_REF] Shahba | Crystal plasticity FE modeling of Ti alloys for a range of strain-rates. Part I: A unified constitutive model and flow rule[END_REF], since the local deformation behavior is predicted more accurately by these methods [START_REF] Busso | Gradient-dependent deformation of two-phase single crystals[END_REF][START_REF] Cheong | Effects of lattice misorientations on strain heterogeneities in FCC polycrystals[END_REF][START_REF] Gao | Mechanism-based strain gradient plasticity-I. Theory[END_REF][START_REF] Fleck | Strain gradient plasticity: Theory and experiment[END_REF][START_REF] Kuhlmannwilsdorf | The theory of dislocation-based crystal plasticity[END_REF][START_REF] Rehrl | Crystal orientation changes: A comparison between a crystal plasticity finite element study and experimental results[END_REF][START_REF] Li | Modeling of deformation behavior and texture evolution in magnesium alloy using the intermediate ϕ -model[END_REF][START_REF] Yao | A polycrystal based numerical investigation on the temperature dependence of slip resistance and texture evolution in magnesium alloy AZ31B[END_REF][START_REF] Zhang | Effect of realistic 3D microstructure in crystal plasticity finite element analysis of polycrystalline Ti-5Al-2.5Sn[END_REF].

With further developing of computational capacity, plenty of image-based models for materials have been created for simulations and property evaluations [START_REF] Thomas | Image-based crystal plasticity FE framework for microstructure dependent properties of Ti-6Al-4V alloys[END_REF][START_REF] Clayton | Dynamic plasticity and fracture in high density polycrystals: constitutive modeling and numerical simulation[END_REF][START_REF] Anahid | Dwell fatigue crack nucleation model based on crystal plasticity finite element simulations of polycrystalline titanium alloys[END_REF][START_REF] Cheng | A crystal plasticity FE model for deformation with twin nucleation in magnesium alloys[END_REF][START_REF] Roters | Crystal Plasticity Finite Element Methods[END_REF][START_REF] Lebensohn | An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials[END_REF][START_REF] Lebensohn | N-site modeling of a 3D viscoplastic polycrystal using Fast Fourier Transform[END_REF][START_REF] Knezevic | Crystal plasticity simulations using discrete Fourier transforms[END_REF][START_REF] Ghosh | Crystal plasticity FE modeling of Ti alloys for a range of strain-rates. Part II: Image-based model with experimental validation[END_REF]. In addition to finite element method(FEM), another robust spectral solver, based on fast Fourier transform(FFT) and Green's function method [START_REF] Lebensohn | An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials[END_REF][START_REF] Lebensohn | N-site modeling of a 3D viscoplastic polycrystal using Fast Fourier Transform[END_REF][START_REF] Knezevic | Crystal plasticity simulations using discrete Fourier transforms[END_REF][START_REF] Ghosh | Crystal plasticity FE modeling of Ti alloys for a range of strain-rates. Part II: Image-based model with experimental validation[END_REF][START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF][START_REF] Moulinec | A fast numerical method for computing the linear and nonlinear mechanical properties of composites[END_REF], was proposed [START_REF] Roters | DAMASK: the Dusseldorf Advanced MAterial Simulation Kit for studying crystal plasticity using an FE based or a spectral numerical solver[END_REF][START_REF] Lee | Modeling the viscoplastic micromechanical response of two-phase materials using Fast Fourier Transforms[END_REF][START_REF] Lebensohn | Fast fourier transform-based modeling for the determination of micromechanical fields in polycrystals[END_REF][START_REF] Kanjarla | Study of internal lattice strain distributions in stainless steel using a full-field elasto-viscoplastic formulation based on fast Fourier transforms[END_REF]. It exhibits more advantages than FEM in dealing with representative volume elements(RVEs) subjected to periodic boundary conditions [START_REF] Eisenlohr | A spectral method solution to crystal elasto-viscoplasticity at finite strains[END_REF][START_REF] Maiti | Fourier-based spectral method solution to finite strain crystal plasticity with free surfaces[END_REF][START_REF] Moore | Crystal plasticity modeling of β phase deformation in Ti-6Al-4V[END_REF].

In this study, we analyzed the relation between microstructure inhomogeneity and fracture behavior with the help of an integrated experimental-simulation method. A realistic microstructure-based RVE was generated to model isotropic CP and tensile deformation simulation using a FFT method. Generally, for Ti alloys, most CP models only consider slip and twinning systems in α phase owing to the limited volume fraction of β phase [START_REF] Thomas | Image-based crystal plasticity FE framework for microstructure dependent properties of Ti-6Al-4V alloys[END_REF][START_REF] Maiti | Fourier-based spectral method solution to finite strain crystal plasticity with free surfaces[END_REF]. However, these models disregard the effect of the β phase on deformation procedure which may lead to discrepancies between experiments and simulations [START_REF] Moore | Crystal plasticity modeling of β phase deformation in Ti-6Al-4V[END_REF]. Thus, both α and β phases were considered in the here proposed model and subsequent simulations. Moreover, post-mortem and fracture surface analyses were performed on a tensile fractured sample. The results were directly compared focusing on the α/β plastic heterogeneities and fracture characteristics.

Experimental Procedure

Sample preparation

The Ti-6Al-4V (TA6V) sample with full lamellar microstructure was used in this work (see Fig. 1(a) and (b) ). The material was supplied by BaoTi Group Co.Ltd and had following chemical composition (in wt%)

6.43Al, 4,21V, 0.40Fe, 0.15Si, 0.10C, 0.05N, 0.015H, and bal. Ti. The β transus temperature of the alloy is 995℃±5℃ [START_REF] Peters | Titanium and Titanium Alloys: Fundamentals and Applications[END_REF]. To obtain a uniform lamellar morphology, the material was treated in the (α+β) phase field.

Subsequently, the material was heated to 960℃ with 1 hour of holding time, followed by air cooling. Next, the material was aged at 540℃ for 8 hours followed by air cooling to ensure an equilibrium distribution of α/β phase.

According to Fig. 1, the considered material exhibits a full lamellar structure before the deformations. Besides, a small quantity of β laths embedded in the α lamellae can be observed. It can be observed that the width of α lamella varies 3 μm~5 μm, whereas β lath seems to maintain the width less than 1 μm (see Fig. 1(c),(d) ).

Fracture surface analysis and post-mortem SEM

In the authors' previous studies [START_REF] Dang | Analysis of shear stress promoting void evolution behavior in an α/β Ti alloy with fully lamellar microstructure[END_REF][START_REF] Dang | Three dimensional X-ray tomographic analysis of crack nucleation and growth by void coalescence in an α/β titanium alloy with lamellar microstructure[END_REF], some tensile tests were completed; the results and details can be found in [START_REF] Dang | Analysis of shear stress promoting void evolution behavior in an α/β Ti alloy with fully lamellar microstructure[END_REF][START_REF] Dang | Three dimensional X-ray tomographic analysis of crack nucleation and growth by void coalescence in an α/β titanium alloy with lamellar microstructure[END_REF]. Only post-mortem fracture surface and cross-sectional microstructure observations were additionally performed here.

The fracture surfaces were characterized via a JSM-6010 scanning electron microscope(SEM). Then, the ruptured samples were embedded into an epoxy matrix with the tensile axis in the plane of the matrix, followed by a routine metallographic sample preparation procedure until the central plane appeared. Finally, polishing and subsequent etching were performed using a mixed solution of OP-S + H 2 O 2 and Kroll's reagent (1 ml HF+ 3ml HNO 3 + 96ml water), respectively. The microstructure morphology and distribution were observed with an optical microscope(OM-Leica DM 4000) and SEM (Zeiss Supra 55VP).

Simulation Procedure

Crystal Plasticity simulation techniques can provide interesting information about the mechanical behavior of polycrystalline materials, based on interplay between phases, grains and sub-grains [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications[END_REF].

To analyze the influence of morphology and distribution of α/β phases on the stress/strain distribution and, subsequently, to predict the vulnerable sites for void nucleation and damage initiation, it is necessary to establish a CP model in accordance with the SEM image. Hereafter, by combining this simulation approach with our previous experiments, the local stress and strain evolution during plastic deformation can be determined.

Isotropic-CP model creation

Based on a realistic SEM image, a two dimensional (2D) FFT model was developed to analyze the distribution of stress/strain in α/β phases with the goal to predict the stress/strain behavior and failure mode of the Ti-6Al-4V alloy.

It is worth to note that, at the ambient temperatures, full lamellar microstructure morphology can be observed. Within one α grain/colony, all α lamellae exhibit parallel and equal crystallographic orientations. To alleviate the effect of the crystallographic orientation on plastic deformation, a local region within an α grain/colony (see Fig. 2 ) can be selected to have α lamellae with the same original crystallographic orientation.

As shown in Fig. 2, the image was imported in Image J [START_REF] Rueden | ImageJ2: ImageJ for the next generation of scientific image data[END_REF][START_REF] Schmid | A high-level 3D visualization API for Java and ImageJ[END_REF] and threshold to distinguish between α phase(dark) and β phase(bright). A code was written to identify and map the phases onto a matrix; one pixel of the image was assigned to one element of the matrix corresponding to the brightness of this pixel in the image (darkness and brightness for α phase and β phase, respectively).

In addition, according to Schmid's law, it is necessary to consider the variations in the Schmid factors in modeling and simulation. The adapted method was as follows,

(1) The original image was rotated 45° and 90°clockwise;

(2) The same region was selected and processed for 0° rotation;

(3) The -image to matrix‖ procedure was repeated.

It should be noted that in this study, the initial RVE region and rotating RVE region were marked as -Ori. 0‖(0° rotation), -Ori. 45‖(45° rotation) and -Ori. 90‖(90° rotation), respectively. At the end of this stage, the map was transferred to the numerical model implemented by Düsseldorf Advanced Material Simulation Kit (DAMASK) spectral solver.

(b) (a) (c)

Isotropic crystal plasticity model

The kinematics of the adapted constitutive model are based on the established continuum mechanical framework of elastoplasticity at finite strains [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications[END_REF].The total deformation gradient F can be decomposed into an elastic part and a plastic part;

ep F = F F (1)
where F e represents the elastic part that consists of elastic stretching and crystal lattice rotation, and F p denotes the plastic deformation gradient due to slip.

The development of the plastic deformation gradient can be expressed as

p p p F = L F  (2)
where L p is the plastic velocity gradient due to dislocation glide and can be described as

     -1 α α p p p L = F F m n   (3)
where   , m α , and n α are shear rate, slip direction vector and normal vector of one slip system α, respectively. n is the number of active slip systems. The shear rate,   ,can be defined as [START_REF] Hutchinson | Bounds and Self-Consistent Estimates for Creep of Polycrystalline Materials[END_REF]  

0 = sgn n S         (4)   =    α α S m n (5)
where 0  is a reference shear rate, τ α , the resolved shear stress of α slip system, and S α , the corresponding critical resolved shear stress. The exponent n represents the stress sensitivity (inverse rate sensitivity) of a dislocation slip.

During plastic deformation, the dislocation accumulation induces the strain hardening. The adapted hardening law is based on Ref. [START_REF] Asaro | Overview no. 42 Texture development and strain hardening in rate dependent polycrystals[END_REF]and Ref.

[50] ,

0 1 a s S S h q S            (6)
where h 0 is the initial hardening parameter, s S  reflects the saturation stress for the α slip system; and q αβ is the latent hardening parameter with a value of 1.4.

As shown in Fig. 2, a selected region was treated as RVE in the CP modeling by using the free and open source package DAMASK [START_REF] Roters | DAMASK: the Dusseldorf Advanced MAterial Simulation Kit for studying crystal plasticity using an FE based or a spectral numerical solver[END_REF].

Since the RVE region was selected within one grain, i.e., misorientation was disregarded in this RVE an isotropic CP model was established in this study.

According to our previous work, no twinning was found during deformation [START_REF] Dang | Analysis of shear stress promoting void evolution behavior in an α/β Ti alloy with fully lamellar microstructure[END_REF][START_REF] Dang | Three dimensional X-ray tomographic analysis of crack nucleation and growth by void coalescence in an α/β titanium alloy with lamellar microstructure[END_REF]. Only slip systems activated in the α/β phases were considered. Therefore, a conventional phenomenological CP model [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications[END_REF][START_REF] Roters | Application of crystal plasticity FEM from single crystal to bulk polycrystal[END_REF], reassessed by Kalidindi et al [START_REF] Kalidindi | Crystallographic texture evolution in bulk deformation processing of FCC metals[END_REF], was implemented for the slip characteristics of hexagonal titanium. The tensile deformation simulations were carried out by a spectral method [START_REF] Shanthraj | Numerically robust spectral methods for crystal plasticity simulations of heterogeneous materials[END_REF] within DAMASK [START_REF] Roters | DAMASK: the Dusseldorf Advanced MAterial Simulation Kit for studying crystal plasticity using an FE based or a spectral numerical solver[END_REF][START_REF] Eisenlohr | A spectral method solution to crystal elasto-viscoplasticity at finite strains[END_REF][START_REF] Shanthraj | Numerically robust spectral methods for crystal plasticity simulations of heterogeneous materials[END_REF].

The identified parameter set is presented in Table 1. The loading rate was set to 10 -3 S -1 , and the direction was parallel to the horizon (X axis).

Results

Microstructure morphology

Fig. 3 depicts the microstructure morphology of the sample that experienced plastic deformation. Adjacent to the fracture, the microstructure seems to maintain a lamellar morphology even after plastic deformation.

A -zig-zag‖ shape fracture trace can be found (marked by yellow dash line) in Fig. 3(a), which infers that this material exhibits a ductile fracture.

In addition, in the local region of the fracture zone, the trace seems to be Furthermore, to analyze the interaction between voids propagation and microstructure morphology, the frontier area, which contains the front edges for void propagation, was selected and marked with a yellow rectangle in Fig. 4(a). The magnified area is presented in Fig. 4(b).

Compared with other areas, the void density in β laths bifurcation area seems to be higher (see white arrows in Fig. 4(b) ). This might be caused because void initiations are favored at the bifurcations of β laths. In addition, approximately all void arrays tend to nucleate and propagate along the α lamellae rather than -breaking‖ the β laths. In other words, the β laths could act as -walls‖ that separate two neighboring arrays. This is further proved by the unbroken β laths, whereas plenty of broken α lamellae can be observed in Fig. 4 (a).

The same results can be noticed in Fig. 4(c) and (d). High density of voids can be found in β bifurcation regions, which means these bifurcation regions are vulnerable sites for void nucleation and propagation.

which can be associated with α lamellae. In Grain 2 and Grain 3, elongated dimples and facets exist. In addition, the dimples in Grain 2 and Grain 3 exhibit different elongation directions, which could be related to different stress/strain distributions owing to dissimilar initial orientations.

Regarding the boundaries, the holes (voids) tend to distribute at the boundary of α grains (grain boundaries are marked by red and yellow dash lines). Typically, a high hole density exists at the triple junction of boundaries (indicated by white arrows). According to Ref. [START_REF] Claudio | Orientation informed nanoindentation of α--titanium: Indentation pileup in hexagonal metals deforming by prismatic slip[END_REF][START_REF] Katani | Micromechanical modelling of damage behaviour of Ti-6Al-4V[END_REF], these holes can be treated as voids. Thus, the grain boundaries or colony boundaries, as well as the junction of boundaries, could be a vulnerable site for void initiation and evolution. In Fig. 5(d), more complex characteristics can be observed; elongated dimples and their precursors-holes (voids), exist (indicated by white arrows) with widths of 5μm~8μm and less than 3μm, respectively. The size of holes(voids) is comparable with the thicknesses of α lamellae(3~5 μm). Within the facets, dimples, holes, β walls and bifurcations exist.

These results seem to agree well with the void distribution in transverse view (see Fig. 4 ).Simultaneously, α fragments exist within the facets in particular at β bifurcations which will be discussed in the following section.

In Fig. 5(e), a tear ridge and the facets can be observed, which may be cleavage features. The tear ridge may result from the boundary of two neighboring α grains with distinct misorientation. Moreover, β walls and bifurcations also exist between facets. Please note that all facets exhibit a plate-like morphology and tend to lie parallel to each other. In addition, the widths of these facets are approximately 5 μm. As mentioned above, the facets could be linked to damaged α lamellae. Further, holes, which could be treated as precursors of dimples, can be found within the facets.

If focusing on the holes, one region was selected (marked by green rectangle in Fig. 5(f) ). Within the facets(linked to α lamellae), the holes tend to distribute at the bottom of β walls. Thus, a void could originate at the interface of α/β phases and propagate along this interface. This assumption agrees well with the results of our previous work [START_REF] Dang | Analysis of shear stress promoting void evolution behavior in an α/β Ti alloy with fully lamellar microstructure[END_REF][START_REF] Dang | Three dimensional X-ray tomographic analysis of crack nucleation and growth by void coalescence in an α/β titanium alloy with lamellar microstructure[END_REF].

Moreover, one micro-crack with holes inside can be observed (pointed out by white arrow in Fig. 5(f) ). This micro-crack can be linked to the coalescence of neighboring void arrays caused by stress/strain concentration; β walls could break during the coalescence. Further, micro-cracks may coalesce to form meso-crack or macro-crack until failure of this material.

CP simulations

According to the microstructure morphology and fractography, the α lamellae and β laths could play different roles during damage evolution.

The α lamellae tend to break to form facets, whereas β lath maintain the original morphology and separate facets (see Fig. 4 and Fig. 5 ). This could be attributed to the heterogeneous stress distribution in the α/β phases, which results from the plastic incompatibility of α/β phases.

In order to analyze the behavior of the α/β phases, a realistic modeling and simulation technique was implemented. In Ref. [START_REF] Katani | Micromechanical modelling of damage behaviour of Ti-6Al-4V[END_REF] and [START_REF] Zhang | Failure mechanism and mode of Ti-6Al-4V alloy under uniaxial tensile loading: Experiments and micromechanical modeling[END_REF], an RVE model for Ti alloy on the basis of real images was established. However, only one α/β phase orientation was considered for polycrystalline material. Therefore, the model disregards the influence of variation in orientation of stress/strain distribution.

Thus, variations in Schmid factors were considered in this current simulation. The results are presented in Fig. 6 created with open source software ParaView [START_REF] Ahrens | 36 -ParaView: An End-User Tool for Large-Data Visualization[END_REF][START_REF] Ayachit | The ParaView Guide: A Parallel Visualization Application[END_REF]. For all initial orientations, heterogeneous stress/strain distributions in the α phase(dark phase) and β phase(bright phase) can be observed. Furthermore, stress concentrates in the regions adjacent to the interface of the α/β phases, in particular in the vicinity of β bifurcations.

Fig. 6 (a)~(c) illustrate the stress and strain map for the Ori.0 status(0° rotation). In the map of components σ xx and σ M (see Fig. 6(a) and (b) ), owing to the plastic-deformation incompatibilities of the α and β phases, the plastic-interaction stresses in the α and β phases are different; the α phase exhibit tensile stress, whereas it is the corresponding compressive interaction stress in the β phase.

These results are in a good agreement with the observations in Ref. [START_REF] Chan | Effects of Deformation-Induced Constraint on High-Cycle Fatigue in Ti Alloys with a Duplex Microstructure[END_REF][START_REF] Chan | Changes in fatigue life mechanism due to soft grains and hard particles[END_REF]. Moreover, in Ref. [START_REF] Chan | Effects of Deformation-Induced Constraint on High-Cycle Fatigue in Ti Alloys with a Duplex Microstructure[END_REF][START_REF] Chan | Changes in fatigue life mechanism due to soft grains and hard particles[END_REF], it was indicated that the crack initiation can be probably promoted by tensile interaction stress. Owing to the plastic-deformation incompatibilities, a high-stress barrier is introduced at interface of α/β phases for slip transmission, which could cause high-density dislocations that pile up along the interface [START_REF] Tan | Slip transmission behavior across α/β interface and strength prediction with a modified rule of mixtures in TC21 titanium alloy[END_REF].

Subsequently, higher strain concentrations at the α/β interface could result in local regions that are favorable for void initiation.

As shown in the strain distribution map (in Fig. 6(c) ), -wrinkles‖ that might be related to the deformation bands exist in α lamellae, with an inclining angle of approximately 45° with respect to the direction of deformation. These observations may reflect the indications in the authors' previous work [START_REF] Dang | Three dimensional X-ray tomographic analysis of crack nucleation and growth by void coalescence in an α/β titanium alloy with lamellar microstructure[END_REF], which investigated the formation of shear bands within one grain. Strain in particular prefers to concentrate at bifurcations of the β phase, which can be interpreted using Ref. [START_REF] Tan | Slip transmission behavior across α/β interface and strength prediction with a modified rule of mixtures in TC21 titanium alloy[END_REF] and [START_REF] Qin | Transgranular shearing introduced brittlement of Ti-5Al-5V-5Mo-3Cr alloy with full lamellar structure at room temperature[END_REF]. Moreover, our results agree well with the observation of a high void density close to bifurcations (shown in Fig. 4(b) and Fig. 5(d) ).

The same results can be found for Ori.45°(45° rotation), as shown in In order to further investigate this fracture behavior, it is necessary to add crystallographic orientation information into the simulation with the help of Electron Backscattered Diffraction technique(EBSD). This work will be conducted in the future.

Discussion

Evidently, void evolution and fracture morphology are related to the microstructure. As shown in Fig. 3(a), a zig-zag shape can be found in the fracture zone. Furthermore, in the fractographs (shown in Fig. 5), cleavage ductile characteristics, holes (precursors of dimples), and α phase fragments within cleavage facets can be observed. Apparently, material ruptured in a hybrid fracture mode.

Hybrid fracture modes have been investigated in Ref. [START_REF] Katani | Micromechanical modelling of damage behaviour of Ti-6Al-4V[END_REF][START_REF] Tan | Slip transmission behavior across α/β interface and strength prediction with a modified rule of mixtures in TC21 titanium alloy[END_REF][START_REF] Qin | Transgranular shearing introduced brittlement of Ti-5Al-5V-5Mo-3Cr alloy with full lamellar structure at room temperature[END_REF][START_REF] Huang | Effect of microstructure on tensile properties of Ti-5Al-5Mo-5V-3Cr-1Zr alloy[END_REF].

However, these studies did not explore the micro-mechanism of this fracture mode intensively.

According to Ref. [START_REF] Faleskog | Micromechanics of coalescence-I. Synergistic effects of elasticity, plastic yielding and multi-size-scale voids[END_REF][START_REF] Helbert | The influence of stress trlaxiality on the damage mechanisms in an equiaxed α/β Ti-6AI-4V alloy[END_REF][START_REF] Tirry | A multi-scale characterization of deformation twins in Ti6Al4V sheet material deformed by simple shear[END_REF][START_REF] Lecarme | Viscoplasticity, damage, and fracture of Ti-6Al-4V[END_REF], voids can nucleate along an α/β interface, owing to a decohesion between the α and β phases. Generally, the β phase could be considered as being composed of -hard‖ particles inserted into a -soft‖ matrix(α phase) [START_REF] Helbert | The influence of stress trlaxiality on the damage mechanisms in an equiaxed α/β Ti-6AI-4V alloy[END_REF][START_REF] Tirry | A multi-scale characterization of deformation twins in Ti6Al4V sheet material deformed by simple shear[END_REF][START_REF] Lecarme | Viscoplasticity, damage, and fracture of Ti-6Al-4V[END_REF]. According to Ref. [START_REF] Mahajan | Surface cracking in α-β titanium alloys under unidirectional loading[END_REF], slip bands originate at the soft α phase near the α/β interface. Subsequently, the soft α phase starts plastic deformation firstly and is more highly strained than the β phase [START_REF] Chan | Effects of Deformation-Induced Constraint on High-Cycle Fatigue in Ti Alloys with a Duplex Microstructure[END_REF][START_REF] Tan | Slip transmission behavior across α/β interface and strength prediction with a modified rule of mixtures in TC21 titanium alloy[END_REF][START_REF] Lecarme | Viscoplasticity, damage, and fracture of Ti-6Al-4V[END_REF][START_REF] Mahajan | Surface cracking in α-β titanium alloys under unidirectional loading[END_REF][START_REF] Hashimoto | The role of elastic interaction stresses on the onset of slip in polycrystalline alpha brass-I. Experimental determination of operating slip systems and qualitative analysis[END_REF][START_REF] Ankem | A rationalization of stress-strain behavior of two-ductile phase alloys[END_REF].

As indicated in Ref. [START_REF] Lecarme | Viscoplasticity, damage, and fracture of Ti-6Al-4V[END_REF], plastic strain accumulates at the interface of the α/β phases and the plastic incompatibility makes the interface a favorable site for void initiation. Besides, the void nucleation is not only driven by plastic strain, but by plastic interaction stress as well, as revealed via FEM simulation.

Based on post-mortem work SEM and FEM simulation for fracture surfaces [START_REF] Lecarme | Viscoplasticity, damage, and fracture of Ti-6Al-4V[END_REF], it was considered that void initiate because of particle/matrix interface decohesion. The position and shape of the voids are depended on the hard particle orientations (perpendicular or parallel to the tensile-stress direction).

These results are in a good agreement with the simulations results of Regarding the observations in Fig. 4 and the fracture surface morphologies(in Fig. 5 ), the cleavage facets exhibit approximately the same morphology and width as the α lamellae(in particular, in

Conclusions

An integrated experimental-numerical method has been implemented to analyze the origin of hybrid damages in the (α+β) Ti-6Al-4V alloy, which exhibits brittle characteristics (cleavage facets) and ductile features (dimples). The following observations have been made:

(1) Owing to microstructure inhomogeneities, the stress/strain tends to distribute heterogeneously in the α/β phases, which could make the interface of the α/β phases a vulnerable site for void nucleation.

(2) Severe stress/strain concentrations in β lath bifurcation areas induce high void densities. In addition, a hard β lath acts as a wall that separates two neighboring void arrays.

(3) The cleavage facets could be linked to ruptured α lamellae caused by a development of voids. The α lamellae can be fractured owing to voids coalescence which completely fill the lamellae.

(4) The orientations of α grains can affect the void evolution by influencing the stress/strain distribution during plastic deformation, This was proved by different fracture surface morphologies for neighboring grains.

Fig. 1 .

 1 Fig.1. Microstructure morphology of Ti-6Al-4V before deformation in different magnifications; (a) 100×, (b) 200×, (c) 500×, (d) 1000×. (Bright lamellae for α phase and dark laths for β phase)

Fig. 2 .

 2 Fig.2. RVE modeling based on realistic image (a) 0°rotation(Ori. 0), (b) 45° rotation(Ori. 45), (c) 90° rotation(Ori. 90)

  Fig.3(b) presents the microstructure morphology of the selected zone

Fig. 3 .

 3 Fig.3. Microstructure morphology of TA6V sample after fractured, (a) low magnification, (b) selected area in (a) (green rectangle), (c) selected area in (a) (red circle)

Fig. 4 .FractographyFig. 5

 45 Fig.4. Microstructure morphology and voids distribution adjacent to fracture zone in (a) and(c); (b) selected region in (a); (d) selected region in (c).

Fig. 5 .

 5 Fig.5. Fractographs of ruptured sample; (a) macro-scale fracture surface morphology, (b), (c) and (e) selected area in (a); (d) and (f) selected area in (c) and (e), respectively.

Fig. 5 (

 5 Fig.5(b) presents the morphology around the junction of α grains or

Fig. 5 (

 5 Fig.5(c) depicts a region selected from Fig.5(a) with cleavage damage

Fig. 6 (

 6 Fig.6(d)~(f). A stress concentration occurs along the α/β interface

Fig. 6 .

 6 Fig.6. Results from the simulations performed using RVE with 5.0% equivalent strain; (a)~(c) for Ori.0; (d)~(f) for Ori.45; (g)~(i) for Ori.90. σ M and ε M refer to von Mises stress and strain , respectively.

this work, shown in Fig. 6 (

 6 Fig.4).

Fig. 5 (

 5 Fig.5(b),(c)and(e) ). Thus, the cleavage facet may be induced by

Table 1

 1 Materials parameters of CP modeling for (a) HCP α phase[START_REF] Levy | Handbook of Elastic Properties of Solids, Liquids, and Gases[END_REF][START_REF] Claudio | Orientation informed nanoindentation of α--titanium: Indentation pileup in hexagonal metals deforming by prismatic slip[END_REF] and (b) BCC β phase[START_REF] Thomas | Image-based crystal plasticity FE framework for microstructure dependent properties of Ti-6Al-4V alloys[END_REF]

	Property	Value	Unit
	(a) α phase (HCP) [54,55]
	C 11	160.0	GPa
	C 12	90.0	GPa
	C 13	66.0	GPa
	C 33	181.7	GPa
	C 44	46.5	GPa
	0 	0.001	S -1
	S 0 for basal slip	349.3	MPa
	S 0 for prism slip	150.0	MPa
	S 0 for pyramidal slip	1107.9	MPa
	S s for basal slip	568.6	MPa
	S s for prism slip	1502.2	MPa
	S s for pyramidal slip	3420.1	MPa
	h 0	200	MPa
	a	2	/
	n	20	/
	(b) β phase (BCC)[11]
	C 11	250	GPa
	C 12	19	GPa
	C 44	230	GPa
	0 	0.0023	S -1
	S 0 ({110}<111> slip system)	500	MPa
	S 0 ( {112}<111> slip system)	500	MPa
	S s ( {110}<111> slip system)	700	MPa
	S s ( {112}<111> slip system)	700	MPa
	h 0	200	MPa
	a	2	/
	n	20	/

Acknowledgement

The authors are grateful for the financial support by International Science and Technology Cooperation Program of China (No.2015DFA51430), National Natural Science Foundation of China (NSFC, No. 51671012, 51671007 and 51401010) and Aeronautical Science Foundation of China (No.2015ZF51069) for this study. The authors would also like to thank Dr. Franz Roters and Dr. Martin Diehl of the Max-Planck-Institutfür Eisenforschung GmbH, for their kind support with the DAMASK code and helpful suggestions for the simulations.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.