Wataru Yoshizaki

Yoshihiko Mochizuki

Naoya Ohnishi

Atsushi Imiya

Free Space Detection from Catadioptric Omnidirectional Images for Visual Navigation using Optical Flow

In this paper, we develop a free space detection algorithm for the visual navigation of the autonomous robot mounting a catadioptiric omnidirectional imaging system. The algorithm detects the dominant plane as the free space from a sequence of omnidirectional images captured by a camera mounted on the autonomous robot. The dominant plane, which can be detected from the optical-flow field, is the largest planar area in the image. For the detection of the dominant plane from the optical-flow field, we adopt the motion separation property, that is, the optical-flow vector is decomposed into infinitesimal rotation, translation, and divergent motions on the images. The algorithm matches the measured translation optical-flow field with the template translation optical-flow field to separate the dominant-plane as the free space for the navigation and the obstacle area. The template optical-flow field is generated from a preobserved image sequence without any calibration of the internal parameters of both the robot and camera.

Introduction

In this paper, we develop the dominant-plane-based [START_REF] Ohnishi | Featureless robot navigation using optical flow[END_REF] free-space-detection algorithm for the autonomous robot mounting a catadioptric omnidirectional imaging system. A featureless and uncalibrated visual navigation for the pinhole camera system is developed by Santos-Victor and Sandini, [START_REF] Santos-Victor | Uncalibrated obstacle detection using normal flow[END_REF] and by Ohnishi and Imiya [START_REF] Ohnishi | Featureless robot navigation using optical flow[END_REF]. The former is a model-based method and the latter is a nonmodel-based method. The latter method detects the free space for the robot navigation using the optical-flow field computed from images captures by the imaging system mounted on the autonomous robot [START_REF] Braillon | Real-time moving obstacle detection using optical flow models[END_REF][START_REF] Bur | Robot navigation by panoramic vision and attention guided features[END_REF], although both methods compute free space from the optical-flow field. The former method generates the model optical-flow field in the pretuning process using images without any obstacles. In this paper, we extend the dominant-plane-detection navigation to the autonomous robot mounting a catadioptric omnidirectional vision system [START_REF] Barreto | Unifying image plane liftings for central catadioptric and dioptric cameras[END_REF][START_REF] Bur | Robot navigation by panoramic vision and attention guided features[END_REF][START_REF] Gaspar | Vision-based navigation and environmental representations with an omnidirectional camera[END_REF][START_REF] Nayar | Catadioptric omnidirectional cameras[END_REF], that is, we use optical-flow field [START_REF] Barron | Performance of optical flow techniques[END_REF][START_REF] Bouguet | Pyramidal implementation of the Lucas Kanade feature tracker description of the algorithm[END_REF][START_REF] Horn | Determining optical flow[END_REF][START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF] on the omnidirectional images for the detection of the dominant plane as the free space for the robot navigation.

For the extension of the previous featureless robot navigation strategy [START_REF] Santos-Victor | Uncalibrated obstacle detection using normal flow[END_REF][START_REF] Ohnishi | Featureless robot navigation using optical flow[END_REF] to the omnidirectional vision system, we adopt the motion separation property, that is, the optical-flow vector is decomposed into infinitesimal rotation, translation, and divergent motions on the images. This geometrical property is well studied on the pinhole camera system. We first clarify the motion separation property of the optical-flow vectors for the omnidirectional images. Then, using this geometrical property of the optical-flow vectors, we develop a free space detection algorithm for the omnidirectional images. In our algorithm, the measured translation optical-flow field of the omnidirectional image is matched with the template translation optical-flow field [START_REF] Santos-Victor | Uncalibrated obstacle detection using normal flow[END_REF], which is generated from a preobserved omnidirectional image sequence, to separate the dominant plane as the free space for the navigation and the obstacle area. Therefore, the output of the algorithm for the navigation is binary images which express the free space and the obstacle area.

The robot navigation in an environment without any landmarks, such as a new environment without any configuration maps for the robot, the detection of the free space, in which the robot can move without any colliding with the obstacles, is a fundamental task. Ohnishi and Imiya [START_REF] Ohnishi | Featureless robot navigation using optical flow[END_REF], Sobey [START_REF] Sobey | Active navigation with a monocular robot[END_REF], Santos-Victor [START_REF] Santos-Victor | Uncalibrated obstacle detection using normal flow[END_REF] and Braillon et al. [START_REF] Braillon | Real-time moving obstacle detection using optical flow models[END_REF] developed an algorithm for the free space detection using optical-flow vectors [START_REF] Sobey | Active navigation with a monocular robot[END_REF][START_REF] Santos-Victor | Uncalibrated obstacle detection using normal flow[END_REF] and a sequence of optical-flow field [START_REF] Ohnishi | Featureless robot navigation using optical flow[END_REF], since the autonomous robot with a vision system automatically detects a sequence of images from the vision system mounted on the robot. The omnidirectional vision system is widely used in the surveillance systems and robot navigation systems [START_REF] Gaspar | Vision-based navigation and environmental representations with an omnidirectional camera[END_REF][START_REF] Lopez-Franco | Omnidirectional vision and invariant theory for robot navigation using conformal geometric algebra[END_REF][START_REF] Winters | Omni-directional vision for robot navigation[END_REF]. Using the wide view of the omnidirectional imaging system, in the surveillance and navigation, the moving objects and the landmarks in the wide area are detected for inspection and robot localisation, respectively. For the robot navigation, the omnidirectional or panoramic views allow the simple computation for localisation of the robot using the several landmarks in the omnidirections detected simultaneously by a single camera. This geometrical advantage is the most important property of the omnidirectional imaging system mounted on the autonomous robot for the navigation and localisation.

In a real environment, the payload of a mobile robot, for example, the power supply, capacity of input devices and computing power, is restricted. Therefore, mobile robots are required to have simple mechanisms and devices [START_REF] Guilherme | Vision for mobile robot navigation: A survey[END_REF][START_REF] Ohnishi | Featureless robot navigation using optical flow[END_REF] for the navigation and localisation. We use an uncalibrated monocular camera as a sensor for obtaining information on the environment. This vision sensor is a low-cost device that is easily mounted on mobile robots. Therefore, we use visual information for the mobile robot navigation.

As same as the pinhole camera system, geometrical features such as lines and planes in the environment are fundamental cues for the panoramic and omnidirectional vision systems to detect the configuration of obstacles in the three-dimensional workspace. If we adopt these traditional strategies, the robot is required to detect the free space as the dual area of the space occupied by obstacle objects. Furthermore, if the map of the workspace is used for the navigation, the robot is required to prepare a geometrical transformation method for transforming the omnidirectional views to the map and reversely to generate omnidirectinal views from the map for the localisation of the robot in the workspace. These two methodologies request for the robot to prepare the special memories for the visual navigation. Therefore, the featureless method developed for the pinhole camera system is suitable for the small payload robot mounting an omnidirectional imaging system, since the algorithm is simple.

Ohnishi and Imiya [START_REF] Ohnishi | Featureless robot navigation using optical flow[END_REF], developed a featureless robot navigation method based on a planar area and an the optical-flow field computed from a pair of successive images. A planar area in the three-dimensional space is called a dominant plane, if it occupies the largest part of an image. We first accept the following four assumptions used by Ohnishi and Imiya [START_REF] Ohnishi | Featureless robot navigation using optical flow[END_REF].

1. The ground plane is the planar area. 2. The camera mounted on a mobile robot is downward-looking. 3. The robot observes the world using the camera mounted on itself for navigation. 4. The camera on the robot captures a sequence of images since the robot is moving.

For the autonomous robot mounting an omni directional camera system, we add the following two assumptions for the camera configuration.

5. The optical axis of the catadioptric omni-directional system mounted on the robot is perpendicular to the ground flow. 6. The planar area occupies more than three fourth of the image.

Although the property of item 5 is the results of the camera-mounting geometry, item 6 is the requirement for the motion separation property on the image. Therefore, if there are no obstacles around the robot and the robot does not touch the obstacles, the ground plane corresponds to the dominant plane in the image observed through the camera mounted on the mobile robot.

Using a vision system inspired by insects, navigation algorithms for the autonomous robot are proposed [START_REF] Franz | Biomimetic robot navigation[END_REF][START_REF] Franz | Insect-inspired estimation of egomotion[END_REF][START_REF] Green | Flying insect inspired vision for autonomous aerial robot maneuvers in near-earth environments[END_REF][START_REF] Sobey | Active navigation with a monocular robot[END_REF][START_REF] Vardy | Biologically plausible visual homing methods based on optical flow techniques[END_REF]. The insect-inspired vision for robot control uses simple information observed by the vision system mounted on the robot. The optical-flow field is a feature computed from an image sequence observed form the moving camera. Therefore, the optical-flow field is a fundamental simple cue for controlling autonomous vehicles [START_REF] Braillon | Real-time moving obstacle detection using optical flow models[END_REF][START_REF] Green | Flying insect inspired vision for autonomous aerial robot maneuvers in near-earth environments[END_REF].

Sobey [START_REF] Sobey | Active navigation with a monocular robot[END_REF] introduced a control strategy for robot navigation using optical flow and the potential field, using the relationship between optical flow and range, inria-00325322, version 1 -28 Sep 2008 which is possibly used by insects for the detection of obstacles in the space for motion control, as shown in Fig. 1. This range in the space detected by the optical flow observed by the camera is used for the generation of the potential, which is used for the computation of the control force to avoid colliding with obstacles. The view from the eyes of flying birds and the compound eyes of insects is a spherical image, which is a standard normalised image for images captured by the omnidirectional vision system. The catadioptric omnidirectional system is widely used in robot vision. In this, paper, we develop an algorithm for the catadioptric imaging system to the detection of free space for the robot navigation without reconstructing three-dimensional geometry of obstacles in the workspace. Therefore, the algorithm is simple and suitable for a low-payload autonomous mobile robot.

Motion Separation and Dominant Plane

Imaging geometry

In this paper, we assume that the catadioptric camera is that with the hyperbolic mirror which is expressed as

- x 2 + y 2 a 2 + z 2 b 2 = 1. ( 1
)
For this hyperbola, setting f z = √ a 2 + b 2 a pair of points P F 1 = (0, 0, f z ) and Geometry of the catadioptric camera system. The robot mounting a catadioptric omnidirectional system moves on the ground floor. The optical axis of the camera system is perpendicular to the ground floor. The robot moves as the combination of the rotation around the gravity axis of the robot and the translation. The focal point (0, 0, fs) of the pinhole camera is located on the focal point of the hyperbola PF .

P F 2 = (0, 0, -f z )
hyperbola z > 0 as the mirror of a catadioptric imaging system, and we locate the focal point of the pin-hole camera on the focal point of the hyperbola (0, 0, -P F ).

For the analysis of the optical-flow vector caused by the rotation and translation of the robot as shown in Fig. 2(a), we adopt the cylindrical coordinate system to express the location of the point P in the space as P = (R cos θ, R sin θ, Z) . In the catadioptric imaging system, the point P in the space is transformed to the point p = (r cos θ, r sin θ) = rn [START_REF] Barreto | Unifying image plane liftings for central catadioptric and dioptric cameras[END_REF] on the imaging plane, for

r = f z z + f z q, q = a 2 -f z γ + √ b 2 (γ 2 + 1) a 2 γ 2 -b 2 , z = qγ + f z , γ = f z -Z R . ( 3
)
This transformation is geometrically shown in Figs. 2(b) and 2(c). Equation [START_REF] Bouguet | Pyramidal implementation of the Lucas Kanade feature tracker description of the algorithm[END_REF] implies that the points P = (R cos θ, R sin θ, Z) and P = (R cos θ, R sin θ, Z ) are imaged to the same point p = (r cos θ, r sin θ) if two points satisfy the con-

dition fz-Z R = fz-Z R .

Optical flow separation

The optical-flow vector is the appearance motion on an image computed from a sequence of images. Optical-flow vectors are the projections of the infinitesimal motion vectors in the space onto the imaging plane. The projection of the infinitesimal motion in the space onto the imaging plane is the optical-flow vector. Assuming that the point P moves to P = P + ∆ during ∆t, we compute the difference of a pair of points p and p on the imaging plane which are the projections of points P and P = (P + ∆), respectively. The optical-flow vector u is computed as

u = ∆p ∆t = p -p = {the image of ∆} = {the image of (P -P )}. ( 4
)
The optical-flow vector is decomposed to the rotation, translation, and divergent vectors as u = u r + u t + u d .

(5)

The rotation vector u r is caused by the rotation of the imaging system around the optical axis of the imaging system. The translation vector u t is caused by the translation of the optical centre parallel to a plane. The divergent vector u d is caused by the motion of the optical centre toward the infinite point in the space.

In this paper, we assume that a catadioptric omnidirectional camera system is mounted on the robot as shown in Fig. 2. The camera observes the vector bundle caused by the translation and rotation of the robot. In this camera configuration, for any camera motion caused by the robot drive, the camera centre does not move in the direction of the optical axis of the camera. Therefore, the camera does not observe the divergent vector bundle as part of the appearance motion.

Rotation optical flow

From eq. ( 3), we have the relation that ∂ ∂θ r = 0. Therefore, if the point P = (R cos θ, R sin θ, Z) moves to the point P = (R cos(θ + ω), R sin(θ + ω), Z) for a small angle ω, the optical-flow vector on the imaging plane is computed as

u(r, θ) = p -p = ωrn ⊥ (6)
for n ⊥ = (-sin θ, cos θ) . Therefore, the optical flow of the rotation of the robot is a collection of co-centred circles on the imaging plane as shown in Figs. 3(a) and 3(c). Furthermore, the speed is independent to the angle θ, that is, ∂ ∂θ |u r | = 0. Moreover, we have the relation for the rotation angular velocity as

ω = 1 r |u r (r, θ)|. ( 7
)
From eq. ( 6), we have the relation

u(r, θ + π) = -u(r, θ). ( 8
)

Translation opticalflow

The optical flow caused by the translation of the camera centre is the image of a line in the space. Assuming that the point P = (R cos θ, R sin θ, Z) moves to

P = (R cos(θ +ω), R sin(θ +ω), Z) with the constant speed v = |∆| = ω D cos 2 θ
in the direction (0, 1, 0) , we set the projections of the pair of points P and P as p = (r cos θ, r sin θ) , p = (r cos(θ + ω), r sin(θ + ω)) .

(

) 9
The pair of points P and P lie on the line (D, D| tan θ|, Z) , -π ≤ θ ≤ π.

From the camera geometry in Figs. 3(b) and 3(d), we have the relations

r = f z f z -Z l -fz(fz-Z) b l + fz(fz-Z) b (3 -2 b 2 a 2 ) | tan θ|, ( 10
)
l = √ D 2 tan 2 θ + (f z -Z) 2 , ( 11
)
r = f z f z -Z l -fz(fz-Z) b l + fz(fz-Z) b (3 -2 b 2 a 2 ) | tan(θ + ω)|, ( 12
)
l = √ D 2 tan 2 (θ + ω) + (f z -Z) 2 . ( 13
)
Using the relation

∆p ∆t = p -p = f z f z -Z    (l - fz (fz -z 1 ) b ) l + fz (fz -Z) b (3-2 b 2 a 2 ) sin(θ + ω) - (l- fz (fz -Z) b ) l+ fz (fz -z 1 ) b (3-2 b 2 a 2 ) sin θ (l - fz (fz -Z) b )| tan(θ+ω)| l + fz (fz -Z) b (3-2 b 2 a 2 ) sin(θ + ω) - (l- fz (fz -Z) b ) tan θ l+ fz (fz -Z) b (3-2 b 2 a 2 ) sin θ    (14)
for R ∼ = R and ω 1, we have the optical-flow vector on the imaging plane as

u t (r, θ) = ωd cos θ ( 1 | tan θ| ) , ( 15
)
where

d = f z f z -Z • l(θ) -fz(fz-Z) b l(θ) + fz(fz-Z) b (3 -2 b 2 a 2 ) , l(θ) = √ D 2 tan 2 θ + (f z -Z) 2 . ( 16
)
From eq. ( 15), we have the relation

u t (r, θ + π) = u t (r, θ), (17)
since l(θ + π) = l(θ). The speed depends on the θ, that is, we have the relation

∂ ∂θ |u t | = 0.

Motion separation

In Fig. 3, (a) shows that the rotation optical-flow vector which is the tangent vectors to the cocentred circles around the optical axis of the catadioptric camera. (b) shows that the translation optical-flow vector is the tangent vectors to the bundle of elliptic arcs which is the projection of the bundle of parallel lines in the space.

From the analysis of the rotation and translation optical-flow vectors, we have the relations

u t (r, θ) = 1 2 u(r, θ) + u(r, θ + π), u r (r, θ) = 1 2 u(r, θ) -u(r, θ + π), ( 18
)
if there is no obstacles in the view. Furthermore, we have the relation for the rotation angular velocity as

ω = 1 2r |u(r, θ) -u(r, θ + π)|. ( 19
)

Dominant plane detection for omnidirectional images

For P = (D tan θ cos θ, D tan θ sin θ, Z) , P = P + ∆, [START_REF] Winters | Omni-directional vision for robot navigation[END_REF] where ∆ = ∆e 2 and e 2 = (0, 1, 0) for a small positive number ∆ and

Q = ( Z -s D tan θ cos θ, Z -c Z D tan θ sin θ, (Z -s) ) , Q = Q + ∆ (21)
where s > 0, setting p and q to be the images of P and Q, respectively, we have the relation,

∆p ∆t < ∆q ∆t . (22)
Therefore, the depth affects to the length of optical-flow vectors. This relation implies that we can discriminate optical-flow vectors on obstacle-region and the ground floor using the difference of the optical flow, if we have the optical-flow vectors on the ground floor.

If obstacles exist, we estimate the rotation velocity as

ω = Median r,θ ( 1 2r |u(r, θ) -u(r, θ + π)| ) (23)
Then, we can separate the translation optical-flow vector of each point as

u t (r, θ) = u -r ωn ⊥ . ( 24
)
Setting u f (r, θ) to be the optical-flow vectors computed from a pair of successive views without any obstacles, we have the relation

|u f (r, θ) -u t (r, θ)| ≤ ε, ( 25
)
for a small positive constant at all points on the ground floor. This relation implies that, setting u f (r, θ) to be the template computed from views without any obstacles, we can detect the dominant plane as

D = {(r cos θ, r sin θ) | |u f (r, θ) -u t (r, θ)| ≤ ε}. ( 26
)

Numerical Results

For the validity evaluation of the method, we have evaluated the detection ratio of the dominant plane. For the qualitative analysis of the results, we used a synthetic environment generated in the computer and a synthetic image sequence observed from the camera mounted on the robot in the synthetic environment. Using 10 synthetic images for each speed, we have evaluated the following measures

E 1 = e1 N × 100[%] and E 2 = e2 N × 100[%]
, where e 1 , e 2 , and N are the area in the free space detected as a part of obstacle area, the area in the obstacle area detected as a part of free space, and the area measure of the view, respectively.

For the computation of the optical-flow vectors on the imaging plane, we adopt the Lucas-Kanade mathod with Pyramid, LKP in abbreviation, for the optical flow computation [START_REF] Bouguet | Pyramidal implementation of the Lucas Kanade feature tracker description of the algorithm[END_REF] in the square grid on the imaging plane. The opticalflow vector decomposition compensates the accurate computation of the opticalflow for the dioptric omnidirectional images. These evaluations show that the estimated ratio is small for the speeds 20mm/s ≤ v ≤ 50mm/s. This result is based on the property of the method as follows -If the speed is too slow, the optical-flow vectors is inaccurately computed for the detection of the dominant plane and obstacles on the ominidirectional images using LKP. -If the speed is too fast, LKP is unstable for the computation of the opticalflow vectors of the catadioptric omnidirectional images. For the generation of the template, we used the area on the ground floor 550mm ≤ R ≤ 2090mm centred at the rotation axis of the catadioptric camera. The geometrical configuration is shown in Fig. 4. In this figure, (a) shows a snapshot of the workspace (b) shows a captured image (c) and (d) show the view fields from the camera mounted on of the robot. (e) and (f) show an observed images for the template optical-flow vector computation and the template optical-flow, respectively. For the computation of the template optical-flow field, we set the translation distance c = 10mm/frame based on the performance evaluation by synthetic data.

Figure 5 shows a results. (a) is an image, (b) shows the optical-flow vectors, (c) the rotation optical-flow vectors, (d) shoes the translation optical-flow vectors, (e) shows the difference between the template optical-flow and translation optical-flow vectors, and (f) is the dominant plane (white area) and obstacles (grey area) detected by the image (e). In the experiment, we set ε = 0.3pixels These results show that our method detects the dominant plane from the catadioptric image sequence observed by the camera mounted on the autonomous robot. This result shows that our method detects the dominant plane as a free space from a catadioptric omni-directional images using optical flow.

Figure 6 shows a comparison evaluation of the method caused by the locations of obstacles in the workspace. In these results, images in the region x ≤ 0 and y ≥ 0 are shown, since the robot is moving in the direction (-1, 0) and the obstacles are located in the left of the robot. These three results show that the obstacles on the both side of the robot and close to the robot are accurately detected.

Conclusions

In this paper, we developed a featureless and uncalibration visual navigation algorithm for the robot mounting a catadioptric omnidirectional imaging system. Using optical flow, our algorithm detects the dominant plane as the free space from a sequence of omnidirectional images captured by an omnidirectional camera mounted on the autonomous robot. For the extension of the previous

Fig. 1 .

 1 Fig. 1. Insect-inspired range navigation. (a)-(c) Insect inspired range potential. An insect understands the range data in the eye-centred coordinates using the optical flow field observed while it is flying around the environment. This range data observed by the moving camera (eye) yields the range-based potential field around the insect for navigation to the local destination. (a) Motion and optical flow. (b) Range data in the eye-centred coordinate. (c) The range-based potential for control to the local destination.In these figures, ẋ, θ, and R are the velocity of the insect, the angular velocity of objects, and the distance to the object, respectively. Then, the relationship θ = ẋ sin θ R is satisfied.

  Fig.2. Geometry of the catadioptric camera system. The robot mounting a catadioptric omnidirectional system moves on the ground floor. The optical axis of the camera system is perpendicular to the ground floor. The robot moves as the combination of the rotation around the gravity axis of the robot and the translation. The focal point (0, 0, fs) of the pinhole camera is located on the focal point of the hyperbola PF .

Fig. 3 .

 3 Fig. 3. The projection of the infinitesimal motion in the space onto the imaging plane is the optical-flow vector. (a) The rotation of the robot causes the rotation opticalflow. The rotation optical flow is the tangent vectors to the cocentred circles around the optical axis of the catadioptric camera. For the rotation, the angle ∠[P PF 1P ] is constant. (b) The translation of the robot causes the translation optical-flow. The translation optical flow is the tangent vector to the bundle of elliptic arcs which is the projection of the bundle of parallel lines in the space. For the translation motion, the length of the line segment P P = c is constant, where |∆| cos θ = ωD cos θ .

Fig. 4 .

 4 Fig. 4. Geometric configuration of robot and template. (a) shows a snapshot of the work space. (b) is a captured image. (c) and (d) show the view fields of the robot. (e) and (f) show an observed image for the template optical-flow field computation and the template optical-flow field, respectively.

Fig. 5 .

 5 Fig. 5. Result. (a)an image, (b)the optical-flow vectors, (c)the rotation optical-flow vectors, (d)the translation optical-flow vectors, (e)difference between the template opticalflow and translation optical-flow vectors, and (f)the dominant plane (white area) and obstacles (grey area) detected by the image (e).

Table 1 .

 1 Errors against speed of the robot τ . For E1 = e 1 N × 100[%] and E2 = e 2 N × 100[%], where e1, e2, and N are the area in the free space detected as a part of obstacle area, the area in the obstacle area detected as a part of free space, and the area measure of the view, respectively.

	Dist.(mm) Thd. = 0.3px Thd. = 0.5px Thd. = 0.7px
		E1	E2	E1	E2	E1	E2
	5	51.6% 0.1% 25.6% 0.3% 13.0% 0.5%
	10	25.9% 0.1% 8.9% 0.4% 4.3% 0.7%
	15	13.8% 0.1% 4.3% 0.4% 2.4% 0.6%
	20	8.1% 0.2% 2.8% 0.5% 1.9% 0.7%
	25	6.2% 0.2% 2.6% 0.4% 1.9% 0.7%
	30	5.5% 0.1% 2.5% 0.5% 1.8% 0.7%
	35	4.8% 0.1% 2.3% 0.4% 1.8% 0.7%
	40	5.1% 0.1% 2.5% 0.5% 1.9% 0.7%
	45	4.8% 0.2% 2.4% 0.4% 1.9% 0.6%
	50	5.2% 0.1% 2.7% 0.4% 2.1% 0.6%
	55	4.8% 0.2% 3.0% 0.4% 2.5% 0.7%
	60	4.7% 0.1% 3.0% 0.4% 2.6% 0.6%

featureless and uncalibarated robot navigation strategies [START_REF] Santos-Victor | Uncalibrated obstacle detection using normal flow[END_REF][START_REF] Ohnishi | Featureless robot navigation using optical flow[END_REF] to the omnidirectional vision system, we adopt the motion separation property, that is, the optical-flow vector is decomposed into infinitesimal rotation, translation, and divergent motions on the images.

Using this geometrical property of the motion vector on the image, we developed an algorithm for the separation of the translation motion and rotation motion from a series of omnidirectional images. The magnitude of the motion vectors for translation motion depends on the range of the obstacles. Therefore, it is possible to separate the appearance motion vectors on an image computed from optical flow to motion on the ground floor and the motion on the obstacles. This property allowed the separation of the free space on an image. Since our algorithm requires template images from translating camera [START_REF] Santos-Victor | Uncalibrated obstacle detection using normal flow[END_REF], we can con-struct template for recognition without any calibration of internal parameters of both the robot and camera. in the pretuning process.


Pengfei Li

Keyi Chen

Fan Wang

Zheng Li

An upper-bound analytical model of blow-out for a shallow tunnel in sand considering the partial failure within the face

Keywords: Shallow shield tunnel, Face stability, Partial failure, Limit analysis, Numerical simulation

This paper proposes a three-dimensional analytical model of blow-out for a shallow tunnel in sand. Upper bound theorem is adopted to deduce the analytical solutions of limit support pressure, and a dimensionless parameter called partial failure ratio is defined to describe the range of partial failure within the tunnel face. A series of numerical simulations are carried out to analyse the limit support pressure and failure zone on the limit conditions. Finally, the analytical solutions including the partial failure ratio and limit support pressure obtained by the proposed model are compared with those obtained by the numerical simulation and the existing global failure model approaches. Results

show that the proposed model in this paper can not only analyze the partial failure with better accuracy, but can also consider the global failure. The proposed model has more extensive applicability in analyzing the face stability of blow-out for a shallow shield tunnel.

Introduction

During the construction of tunnel engineering, the excavation may weaken the tunnel face and cause face failure, especially for shallow tunnels [START_REF] Yoo | Finite-element analysis of tunnel face reinforced by longitudinal pipes[END_REF][START_REF] Khezri | The stability of shallow circular tunnels in soil considering variations in cohesion with depth[END_REF][START_REF] Patil | Behavior of shallow tunnel in soft soil under seismic conditions[END_REF][START_REF] Ding | Face stability analysis of shallow circular tunnels driven by a pressurized shield in purely cohesive soils under undrained conditions[END_REF]. Tunnel excavation mainly refers to the closed-face case, like the open-face conventional tunneling or the shield tunneling [START_REF] Pan | Safety factor assessment of a tunnel face reinforced by horizontal dowels[END_REF]. Conventional tunneling does not exert support pressure on tunnel face, but developing some auxiliary techniques, such as advance injection tubes, shed-pipe grouting, face bolts and umbrella arch system [START_REF] Ocak | Control of surface settlements with umbrella arch method in second stage excavations of Istanbul Metro[END_REF][START_REF] Aksoy | The role of umbrella arch and face bolt as deformation preventing support system in preventing building damages[END_REF][START_REF] Maidl | Handbook of Tunnel Engineering[END_REF][START_REF] Oke | Numerical analyses in the design of umbrella arch systems[END_REF][START_REF] Zhang | Load transferring mechanism of pipe umbrella support in shallow-buried tunnels[END_REF][START_REF] Anagnostou | Analysis method and design charts for bolt reinforcement of the tunnel face in cohesive-frictional soils[END_REF]Paternesi et al. 2017). These techniques generally improve the face stability by limiting the excessive deformation of the surrounding stratum, or reinforcing the rock or soil mass of the tunnel face.

Shield tunneling method has been more and more widely adopted in the construction of subway tunnels, for its advantages of higher security and slighter effect on complex urban environments.

During the construction of shield tunnels, the face stability is a key issue to ensure the engineering safety [START_REF] Liao | Shield tunneling and environment protection in Shanghai soft ground[END_REF][START_REF] Sterpi | Soil nailing at the tunnel face in difficult conditions: a case study[END_REF][START_REF] Jing | A case study of TBM performance prediction using field tunnelling tests in limestone strata[END_REF]. The chamber pressure is an effective support pressure to balance the water and soil pressure in front of tunnel face, and greatly influences the tunnel face stability [START_REF] Zhou | PSO-based Elman neural network model for predictive control of air chamber pressure in slurry shield tunneling under Yangtze River[END_REF], Chen et al., 2018). If the applied chamber pressure is not sufficient to resist against the pressure from the surcharge and soil self-weight, active failure (collapse) occurs. On the other hand, blow-out, also referred to as passive failure, appears if the applied chamber pressure is high enough to 'push' the soil towards the ground surface [START_REF] Klotz | A 3D finite element simulation of a shield tunnel in weathered Singapore Bukit Timah Granite[END_REF][START_REF] Mollon | Rotational failure mechanisms for the face stability analysis of tunnels driven by a pressurized shield[END_REF]. In practical engineering, the blow-out, which is triggered by the high chamber pressure, is very common in shallow shield tunnels and results in huge economic losses. Therefore, researches on blow-out of shield tunnel face are worthy of carrying out.

Scholars have investigated the tunnel face stability mainly through three approaches: theoretical analysis, experimental test and numerical simulation. However, most of the researches were just available for the collapse failure [START_REF] Broms | Stability of clay in vertical openings[END_REF][START_REF] Davis | The stability of shallow tunnels and underground openings in cohesive material[END_REF]Murayama, 1996;[START_REF] Subrin | Tunnel face stability in frictional material: a new 3D failure mechanism[END_REF][START_REF] Fang | Shallow tunnelling method (STM) for subway station construction in soft ground[END_REF][START_REF] Fang | Effects of twin tunnels construction beneath existing shield-driven twin tunnels[END_REF]. There were few studies on the blow-out, that is, the blow-out in front of the tunnel face. As for the theoretical analysis, a reasonable failure mode of soil can be used to describe more accurately the limit support pressure. Thus, the assumption of the failure model in front of tunnel face has been widely taken into account to analyze the tunnel face stability. [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF] proposed a blow-out mechanism composed of a single conical block outcropping at the ground surface for very shallow tunnels in weak soils. Based on the model of Leca and Domreiux (1990), [START_REF] Soubra | Three-dimensional face stability analysis of shallow circular tunnels[END_REF] improved the single-block into a new blow-out failure mechanism consisting of two truncated rigid cones and a log spiral, and the relationship between the cohesion coefficient N c and the surcharge coefficient N s was given. Then, [START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF] improved the failure region above into five truncated rigid cones, which further optimized the results of the upper bound solution. [START_REF] Mollon | Rotational failure mechanisms for the face stability analysis of tunnels driven by a pressurized shield[END_REF] employed the spatial discretization technique and presented a rotational failure mechanism delimited by two logarithmic spirals in the central vertical plane of the tunnel, which made it possible to involve the entire circular tunnel face. [START_REF] Senent | Tunnel face stability in heavily fractured rock masses that follow the Hoek-Brown failure criterion[END_REF] used limit analysis method and nonlinear Hoek-brown failure criterion respectively to investigate the failure of tunnel face and crown. Chen et al. (2013) studied the blow-out failure mode under the sandy soil layer and modified the prism in the traditional three-dimensional wedge model to a chamfered platform with a certain inclination angle. Compared to the limit equilibrium methods [START_REF] Horn | Horizontal earth pressure on the vertical surfaces of the tunnel tubes[END_REF][START_REF] Jancsecz | Face support for a large mix-shield in heterogeneous ground conditions[END_REF][START_REF] Anagnostou | Face stability condition with earth pressure balanced shields[END_REF][START_REF] Broere | Tunnel face stability and new CPT applications[END_REF], the limit analysis methods have a strictly theoretical basis and provide better results [START_REF] Han | Upper-bound solutions for the face stability of a shield tunnel in multilayered cohesive-frictional soils[END_REF]. In addition, the numerical approach [START_REF] Li | Stability analysis of large slurry shield-driven tunnel in soft clay[END_REF][START_REF] Wong | Three-dimensional analysis of passive failure of circular tunnel face in layered ground[END_REF] and experimental approach [START_REF] Li | Laboratory study on face stability mechanism of slurry shields[END_REF]Wong, 2012) were also widely used in the analysis of blow-out.

At present, it has been concluded from some researches that there exists a gradual evolution process from partial failure to global failure within the tunnel face when blow-out caused by the excessive support pressure occurs. Global failure means that the failure range within the tunnel face covers the entire tunnel face, while partial failure area includes only the upper part of the tunnel face. [START_REF] Dias | Three-Dimensional Face Stability Analysis of Circular Tunnels by Numerical Simulations[END_REF] performed a series of numerical simulation by FLAC 3D for a shallow tunnel and concluded that the partial failure rather than global failure was firstly observed within the tunnel face when the blow-out occurs. [START_REF] Li | Stability analysis of large slurry shield-driven tunnel in soft clay[END_REF] obtained the displacement contour of tunnel face in blowout for the Shanghai Yangtze River Tunnel by numerical simulation. It was shown that the conspicuous displacement occurs in the upper part of the tunnel face, while the displacement was hardly observed near the tunnel invert. Thus, the partial failure within the tunnel face of shallow tunnels is a significant problem that needs to pay much more attention. However, most of the existing researches for tunnel face stability all focused on the global failure within the whole tunnel face. The analytical solution for partial failure in blow-out was seldom studied.

This paper aims at proposing an analytical model of blow-out for a shallow tunnel in sand under drained condition to analyse the partial failure within the tunnel face. The analytical solution of limit support pressure is derived based on the upper-bound theorem. Then, a series of numerical simulations are performed to obtain the limit support pressure and failure zone in front of the tunnel face. Finally, the analytical solutions including the partial failure ratio and limit support pressure obtained by the proposed model are compared with those obtained by the numerical simulation and existing approaches.

Proposed model

Problem description

Based on the single conical block mechanism proposed by [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF]Dormieux (1990), Soubra et al. (2008) presented a translational kinematically admissible failure mechanisms composed of several truncated rigid cones. These rigid cones with an opening angle 2φ are reversed and the mechanism always outcrops. Although this model has been widely used to analyze the blow-out case, it is not capable to the analysis of partial failure within the tunnel face. Therefore, as indicated in Fig. 1, a failure model is established to analyse the partial failure in blow-out within the shallow tunnel face in dry sand. This model further improves the model proposed by [START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF]. Only five blocks are necessary for the blow-out mechanism since an increase in the number of blocks improves the solutions by a few percent. The soil is assumed to be homogeneous, isotropic and characterized by the Mohr-Coulomb yield criterion.

As shown in Fig. 1, a tunnel of diameter D excavated under a cover depth C is considered. A surcharge σ s is applied on the ground surface, and a uniform pressure σ T is imposed on the tunnel face [START_REF] Ibrahim | Three-dimensional face stability analysis of pressurized tunnels driven in a multilayered purely frictional medium[END_REF][START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF], Li et al., 2009, Mollon et al., 2013, Soubra et al., 2008).

Meanwhile, a dimensionless parameter called partial failure ratio (η=d/D) is defined to describe the range of partial failure within the tunnel face. Specifically, the blow-out case with global failure can be taken into account when η=1 is applied; when 0<η<1, the blow-out case with partial failure occurs with a partial failure ratio η.

Analytical solutions

Geometric properties

(1) Angle parameters According to the geometric properties, α i (for 1 ≤ i ≤ 5) can be given by ( 1)

1 = + 2      (2) 2 1 = + + 2       (3) 3 1 2 = + + 2         (4) 4 1 2 3 = + + + 2          (5) 5 1 2 3 4 = + + + 2           
(2) Length parameters

The intersection of the first block and tunnel face is an ellipse with a long axis of AB 1 in vertical direction. It can be known that:

(6) 1

AB d 

The intersections of adjacent blocks are also ellipses with a long axis of AB i that are calculated as:

(i = 2, 3, 4, 5) (7) 1 1 1 1 sin sin( ) i i i i i AB AB         
The busbar of the truncated cone AB i O i is

(i = 1, 2, 3, 4, 5) (8) sin sin 2 i i i AO AB   
The height of the truncated cone AB i O i and EFO 5 is obtained by

(i = 1, 2, 3, 4, 5) (9)
sin( 2)

i i i h AO     (10) 4 6 5 5 1 h cos( 2 ) i i C AO         
(3) Area and volume parameters

The area A 1 of the ellipse which intersects between the first block and tunnel face is as follows:

(11) 2 2 1 2 sin 1 4 cos d A     
In addition, the area A i of the ellipse which intersects between the two adjacent blocks is described as:

(i = 2, 3, 4, 5) (12) 2 2 2 sin 1 4 cos i i i AB A     
The elliptical area A 6 where the last truncated cone intersects the ground is denoted as:

(13) 2 2 6 6 2 sin 1 4 cos EF A     
The volume (for 1 ≤ i ≤ 5) and lateral area S i (for 1 ≤ i ≤ 5) of the rigid truncated elliptical cone i are expressed as:

(i = 1, 2, 3, 4, 5) (14) 1 1 1 ( ) 3 i i i i i V A h A h     (15) 2 1 2 1 cos cos sin sin S A A       (i = 2, 3, 4, 5) (16) 1 1 cos cos sin sin i i i i i S A A         (4) Related characteristic parameters (17) 1    (i = 2, 3, 4, 5) (18) 1 1 i i i        (19) 4 6 5 1 i i          (20) 5 4 4 5 5 1 1 sin 2 ( ) cos( 2 ) sin( + ) 2 i i i i C EF AO                
(5) Velocity relationship

The velocity hodograph is given in Fig. 2. The velocities relationships are determined as follows:

(i =1, 2, 3, 4) (21) 1 1 sin( 2 ) sin( 2 ) i i i i i              (i =1, 2, 3, 4) (22) 1 , 1 1 sin( ) sin( 2 ) i i i i i i i              
The angle ψ i (for 1 ≤ i ≤ 5) between velocity v i (for 1 ≤ i ≤ 5) and vertical direction are obtained as follows:

(23)

1 1      (i = 2, 3, 4, 5) (24) 1 1 i i i i i           2.2.2. Limit support pressure
The power Pσ T of the support pressure, the power Pσ s of the surcharge, the power P γ of the soil unit weight and the power P e of external loads can be calculated by Eqs. (25~28)

(25) 1 1 1 A sin( ) T T P        (26) 6 5 5 A cos s s P       (27) 4 5 5 5 1 [ cos cos ] i i i i P V V            ( 28
)
T s e P P P P

     
The relationship between the dissipation power P v and cohesion c of the blow-out region is denoted as

(29) 4 4 5 5 1 ,i 1 1 1 [ S S ]cos i i i i i i P c A              
According to the upper bound theorem of the limit analysis method, the condition for the stability of the tunnel face is 25), ( 28) and ( 30), the limit support pressure is denoted as

(31) 1 1 1 A sin( ) s T P P P           
To be more concise, the support pressure function of the model is determined as follows:

(32)

T c s s N D cN N        
where N s , N γ and N c are used as non-dimensional coefficients to indicate the influence of surcharge, the soil weight and cohesion on the limit support pressure, respectively.

(33)

6 5 2 4 1 1 sin(2 2 ) cos s A N A            (34) ( 1) tan s c N N    (35) 1 2 3 4 5 1 ( ) A cos N N N N N N D        (36) 1 1 sin N V   (37) 2 2 2 1 1 sin(2 ) N V         (38) 3 3 3 2 1 sin(2 ) N V         (39) 4 4 4 1 3 1 sin(2 2 ) N V           (40) 5 5 5 2 4 1 sin(2 2 ) N V          
From the equations above, it is found that the limit support pressure is determined by the limit partial failure ratio η, the angle parameters α and β i (i =1, 2, 3, 4). The optimal solution of limit support pressure can be obtained by a mathematically.

Numerical simulations by using FLAC 3D

Numerical models

The analysis above shows that the factors affecting the limit support pressure consist of tunnel covered depth C, tunnel diameter D and friction angle φ. A series of numerical models performed by the program FLAC 3D are established to analyse the influence of these factors and verify the analytical model. Twelve cases, as shown in Table 1, are simulated. For the sake of simplicity, it is assumed that the soil is cohesionless (c=0) and its unit weight γ is 18 kN/m³. Also, the surcharge is assumed to be zero. In this section, numerical analyses were conducted by a finite difference computer program FLAC 3D are reviewed to calculate limit passive pressure and verify the partial failure proposed in this paper. As shown in Fig. 3, considering the symmetry, the model shows only half of the circular tunnel cut longitudinally along the central axis. The boundary conditions of the model are given as follows:

the ground surface is free; the lateral displacement boundaries are fixed in the normal direction and the displacement boundaries at the bottom are fixed in both the horizontal and vertical directions [START_REF] Li | Numerical parametric study on stability and deformation of tunnel face reinforced with face bolts[END_REF]. The longitudinal length of the model is 80 m and the transverse length is 5 times tunnel diameter (5D). The total height of the model is equal to C+6D. The three-dimensional model were determined by using non-uniform grids, including 104 871 nodes and 100 320 elements (for C/D=1.0).

The model of the numerical simulation is characterized by a Mohr-Coulomb yield criterion. The Poisson's ratio of soil layer is set as 0.35 and the bulk modulus is set as 80 MPa.

The detailed steps of numerical simulation are described as follows:

(1) Establishing the original sand layer model.

(2) A simplified single-step excavation scheme, which has been successfully adopted in many previous researches [START_REF] Gioda | Developments and applications of the numerical analysis of tunnels in continuous media[END_REF], is simulated to focus on the analysis of tunnel face stability. In this paper, 30 m is excavated in one step when performing the numerical simulation (see Fig. 3). At the same time, the fixed constraints are imposed on each node around the tunnel perimeter that has been excavated.

(3) Then the support pressure, which equal to the initial ground horizontal stress for the center of the tunnel, is applied on the tunnel face. By increasing the support pressure applied on the tunnel face gradually, the relationship curves between the support pressure and the horizontal displacement of the central point within the tunnel face can be plotted.

(4) When the support pressure applied on tunnel face increases to a certain value, the horizontal displacement of the central point within the tunnel face increases sharply. At this moment, the limit support pressure for the blow-out case is obtained.

Numerical results

Limit support pressure

Fig. 4 shows the relationship between the support pressure ratio P/P 0 (the ratio of the specified face support pressure to the initial ground horizontal stress for the center of the tunnel) and the horizontal displacement of the central point within tunnel face for the 12 cases are shown in Table 1.

It can be seen that the horizontal displacement of the central point within the tunnel face increases with the increase of the support pressure P. When P increase to a certain value, the horizontal displacement of the central point does not converge and increases dramatically. At this time, the soil is considered to reach the limit state and the support pressure P is expressed as σ T . As shown in Fig. 4(a), the values of σ T /P 0 increase from 3.31 to 5.80 with the increase of buried depth C in the case of the same other conditions. Similarly, from Fig. 4(b), it can be seen that with the increase of diameter D, the values of σ T /P 0 increase from 1.70 to 2.76, while those increase from 5.34 to 21.60 with the increase of friction angle φ in Fig. 4(c). Therefore, the deeper depth C, the smaller tunnel diameter D and the higher friction angle φ, corresponding to greater limit support pressure ratios. Moreover, it appears that the influence of tunnel depth and friction angle on the limit support pressure ratios is more obvious. The following discussion will focus on the effects of the three factors of the cover depth C, the tunnel diameter D and the friction angle φ on partial failure ratio and the limit support pressure. Support pressure ratio ( P / P 0 )

Horizontal displacement of the central point within tunnel face (m) That is to say, the partial failure occurs within the tunnel face for Case 4 and Case 5. Fig. 6 shows the displacement contours in limit conditions for Case 9, Case 2, Case 4 and Case5.

For Case 9 (C=3m, D=6m, φ=20°), as observed in Fig. 6(a), the values of the isoline greater than 0.15m change uniformly, while the values of the contour line less than 0.15m change significantly.

According to the analysis of [START_REF] Zhang | Face stability analysis of shallow circular tunnels in cohesive-frictional soils[END_REF], "the position within a sudden increase gradient can be defined as the boundary strip of the failure zone." If the isoline with displacement value of 0.15m is taken as the boundary strip of partial failure, the distance from the boundary strip to tunnel crown within tunnel face is about 3.6m. This value is approximately consistent with the result obtained from Note that, the rule between η and D described in Fig. 9

Effect of lateral pressure coefficient (K 0 )

The lateral pressure coefficient K 0 , which is crucial to estimate the initial stresses, has a great influence on the stability of shallow tunnels (Chen et al. 2013;[START_REF] Lei | Calculation of the surrounding rock pressure on a shallow buried tunnel using linear and nonlinear failure criteria[END_REF]Qarmout et al. 2019).

Fig. 12 gives the relationship between the partial failure ratio η and the lateral pressure coefficient K 0 . The values of soil parameters, tunnel diameter and cover depth are the same as case 9. According to [START_REF] Golpasand | Effect of the lateral earth pressure coefficient on settlements during mechanized tunneling[END_REF], the values of K 0 are in the range from 0.5 to 3.0. Note that, since the limit analysis method does not take the K 0 parameter into account, the theoretical results are constant and do not vary with the values of K 0 . In addition, η obtained by numerical simulation does not change with the increase of K 0 , basically. Moreover, the theoretical solutions match well with the numerical solutions.

Effect of surcharge (σ s )

The analysis of cover depth, tunnel diameter, friction angle and lateral pressure coefficient in the above sections is carried out under the assumption that the ground surcharge is zero. However, when tunneling below ground surface with existing surcharge, the movement and deformation of the stratum are different from those with stress free ground surface [START_REF] Yamamoto | Stability of a circular tunnel in cohesive-frictional soil subjected to surcharge loading[END_REF][START_REF] Ali | Undrained stability of a single circular tunnel in spatially variable soil subjected to surcharge loading[END_REF][START_REF] Wang | The analytical predictions on displacement and stress around shallow tunnels subjected to surcharge loadings[END_REF]. Fig. 13 indicates the variation of partial failure ratio η as the increase of ground surcharge σ s . The values of soil parameters, tunnel diameter and cover depth are the same as case 9. As can be seen, the values of η derived from the proposed model increase to a constant value and then remain unchanged (i.e. η=1.0) as ground surcharge σ s further increases, while the numerical results increase relatively slowly.

Limit support pressure

Effect of cover depth (C)

For a certain value of D, Fig. 14 shows the comparison of limit support pressure between the proposed model, numerical simulation and existing research results [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF][START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF][START_REF] Ocak | Control of surface settlements with umbrella arch method in second stage excavations of Istanbul Metro[END_REF] in the sand with a small friction angle (φ=15°). In Fig. 14, the limit support pressures in blowout obtained by the theoretical model and numerical simulation increase with the increase of C.

Combined with the analysis of Fig. 8, when the cover depth C is less than the critical cover depth, the η is less than 1.0 and the partial failure of the face occurs. When the cover depth C reaches or exceeds the critical cover depth, the partial failure ratio η is equal to 1.0, that is, the failure mechanism of tunnel face evolves from partial failure to global failure. Note that, the upper bound theorem provides an unsafe estimate of the support pressure required to maintain face stability (higher or equal to that actually required for blow-out) (Mollon et.al 2010). Therefore, a smaller limit support pressure indicates a better solution. In case of C=3m (C/D=0.5, φ=15°), the value of σ T reduced by 7.2% with respect to the results of [START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF], and the reduction attains 14% relative to the results presented by [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF]. The comparison results illustrate that the proposed model is suitable for calculating the case of partial failure. When C=6m (C/D=1.0), partial failure ratio η is equal to 1.0 and the limit support pressure in this paper is in agreement with that obtained by global failure model by [START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF].The above analysis demonstrate that the proposed model in this paper not only can analyze the partial failure with better accuracy, but also can consider the situation of the global failure (η=1), which indicates that the model has more extensive applicability.

Effect of tunnel diameter (D)

In the case of C=3m, as tunnel diameter D increases, Fig. 15 gives the results of limit support pressure provided by the proposed model and the existing approaches. It can be seen that the differences of results between the global failure model [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF][START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF][START_REF] Ocak | Control of surface settlements with umbrella arch method in second stage excavations of Istanbul Metro[END_REF] and partial failure model are obvious in Fig. 15. The limit support pressure σ T of the proposed model based on the assumption of partial failure remains unchanged, while that of the global failure model proposed by [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF] and [START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF] increases linearly. As shown in Fig. 15, the results of σ T obtained by the proposed model is the closest to those obtained by numerical simulation.

In addition, the reduction with respect to the results provided by [START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF] and [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF] can attain 41.2% and 42.9% when D=16m (C=3m, φ=15°) , respectively. Therefore, it can be concluded that the partial failure in blow-out is more common than the global failure for the largediameter shallow tunnels, which has also proved in the literature of [START_REF] Li | Stability analysis of large slurry shield-driven tunnel in soft clay[END_REF]. For a constant value of C/D, the theoretical models mentioned in Fig. 16 demonstrate the same variation rule. The limit support pressure σ T increases linearly as the tunnel diameter D increases.

Compared to the other theoretical models shown in Fig. 16, the results obtained by the proposed model gives better results since the present upper-bound solutions are smaller. For example, when D=12m (C/D=0.5, φ=15°), relative to the result presented by [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF] and [START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF], the value of σ T are improved (reduced) by 14% and 7.2%, respectively.

Effect of frictional angel (φ)

In the case where C/D is constant,Fig. 17 presents the comparison of limit support pressure in various friction angle between the proposed model, numerical simulation and the existing approaches.

It appears that the limit support pressure σ T increase gradually with the increase of the friction angle regardless of the proposed model, the numerical simulation and the existing approaches. In Fig. 17, the single conical block model proposed by [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF] is more sensitive to φ and its values of σ T are larger than those of other mentioned models. In addition, compared with the multi-block model presented by [START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF], the limit support pressure obtained by the proposed model is smaller and closer to the results obtained by numerical simulation. Therefore, the limit support pressures obtained by the proposed model is improved. In the case of φ=20°(C=3m, D=6m), the improvement (reduction) with respect to the results by [START_REF] Leca | Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material[END_REF] and [START_REF] Soubra | Three-dimensional face stability analysis of circular tunnels by a kinematical approach[END_REF] are 15.2% and 4.8%, respectively. However, Fig. 17 also shows that the deviation between the theoretical solution and the numerical solution becomes greater as the friction angle increases. 4.2.4 Effect of lateral pressure coefficient (K 0 )

The relationship between the limit support pressure ratio P/P 0 and lateral pressure coefficient K 0 is shown in Fig. 18(a). It seems that P/P 0 decreases with the increase of K 0 . Fig. 19 demonstrates the horizontal displacement curves within tunnel face in the limit state for various lateral pressure coefficient (i.e. K 0 =0.5, 1.0, 1.5, 2.0, 2.5, 3.0).The horizontal displacement within tunnel face are greatly affected by the values of K 0 . When K 0 =0.5, a relatively large displacement can be observed. With the increases of K 0 , the horizontal displacement gradually decreases. Subsequently, the horizontal displacement increases as K 0 further increases. This variation can be more intuitively reflected in Fig. 20, which describes the relationship between the lateral pressure coefficient and the maximum displacement within tunnel face. In summary, the lateral pressure coefficient has a great influence on the displacement, but has almost no influence on the limit support pressure, which was also presented by [START_REF] Vermeer | Tunnel heading stability in drained ground[END_REF].

K 0 =0.5 K 0 =1.0 K 0 =1.5 K 0 =2.0 K 0 =2.5 K 0 =3.0

Conclusions

The paper presents an analytical model of blow-out for shallow tunnels in sand under drained condition considering the partial failure within tunnel face based on the upper bound theorem. A series of numerical simulations are carried out to analyse the influence of the cover depth, tunnel diameter and the friction angle on the partial failure ratio and limit support pressure. The main conclusions are as follows:

(1)Compared with the classical solutions, the analysis shows that the proposed model gives better results in solving the passive partial failure ratio and the limit support pressure.

(2)The cover depth of the tunnel is of great importance to the partial failure ratio and limit support pressure. The greater the cover depth ratio, the greater the partial failure ratio. When the cover depth ratio exceeds 1.0, the failure mechanism of tunnel face is no longer the partial failure, but shows the characteristic of global failure. Under the condition of constant tunnel diameter, the limit support pressure increases with the increase of the cover depth.

(3)Contrary to the cover depth, the tunnel diameter has little effect on the range of blow-out area and the limit support pressure of tunnel face. The blow-out area mainly occurs in the upper part of tunnel face. In the case of the same cover depth, the larger the tunnel diameter, the smaller the partial failure ratio, while the limit support pressure tends to be constant.

(4)It is worthwhile to note that friction angle is critical for the partial failure ratio and limit support pressure. In the limit state, as the friction angle increases, the failure range of tunnel face expands continuously, and the maximum outcrop point of the ground approaches toward the tunnel face. Meanwhile, the limit support pressure continues to increase. However, the deviation between the proposed model and the numerical simulation also becomes more obvious with the friction angle increases, which needs further investigations.

(5)The lateral pressure coefficient is one of the most important parameters in geotechnical engineering. However, it affects the deformation and the limit support pressure ratio within the face rather than the limit support pressure. As the lateral pressure coefficient increases, the limit support pressure ratio decreases gradually, while the deformation within the face decreases at first and then increases. Besides, it seems that the change of lateral pressure coefficient does not have a remarkable effect on the partial failure ratio.

(6)The ground surcharge affects both the deformation of the stratum and tunnel face stability.

The limit support pressure increases linearly with the increase of ground surcharge, which means the value of surcharge greatly affects the tunnel face stability. In addition, the partial failure ratio increases at first with the increase of ground surcharge and then tends to be constant.

Fig. 1 .

 1 Fig. 1. The new 3D partial failure mechanism.

Fig. 2 .

 2 Fig. 2. The velocity field of the new 3D failure mechanism.

Fig. 3 .

 3 Fig. 3. Numerical model.

Fig. 4 .

 4 Fig. 4. Relationship between P/P 0 and horizontal displacement.3.2.2. Failure zone on the limit conditions

Fig. 5 .

 5 Fig. 5. Horizontal displacement curve of tunnel face obtained by numerical simulation.

Fig. 5 Fig. 6 .

 56 Fig. 5 (a). The comparison is also shown in Fig. 6(a). The red line represents the boundary of the failure zone provided by numerical simulation, and the black line represents the failure zone boundary

Fig. 7 Fig. 7 .Fig. 8 .

 778 Fig.7shows the displacement of ground surface in limit conditions for various friction angles

Fig. 9 .

 9 Fig. 9. Effects of D onη and d (C=3m, φ=15°).

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Effects of D on η (C/D=0.5, φ=15°).4.1.3. Effect of frictional angel (φ)For a certain value of C/D, the relationship between the partial failure ratio η and the friction angle φ is shown in Fig.11. It can be seen that the partial failure ratio η obtained by the proposed model first increases and then decreases gradually, while that obtained by the numerical simulation shows an increasing trend as a whole. When φ<30°, the values of η obtained by the proposed model are close to those obtained by the numerical simulation. However, the relatively large deviation occurs in the case of larger friction angle (φ>30°), which needs further investigation.

Fig. 12 .

 12 Fig. 12. Effects of K 0 on η (C=3m, D=6m, φ=20°).

Fig. 13 .

 13 Fig. 13. Effects of σ s on η (C=3m, D=6m, φ=20°).

Fig. 14 .

 14 Fig. 14. Effects of C on σ T (D=6m, φ=15°).

Fig. 15 .

 15 Fig. 15. Comparison of σ T with the variation of Tunnel diameter (C=3m, φ=15°).

Fig. 16 .

 16 Fig. 16. Effects of D on σ T (C/D=0.5, φ=15°).

Fig. 17 .

 17 Fig. 17. Comparison of σ T with the variation of friction (C=3m, D=6m).

Fig. 18 .

 18 Fig. 18. Effects of K 0 on P/P0 and σ T (C=3m, D=6m, φ=20°).

Fig. 19 .

 19 Fig. 19. of K 0 on horizontal deformation (C=3m, D=6m, φ=20°).

Fig. 20 .

 20 Fig. 20. Maximum displacement within tunnel face (C=3m, D=6m, φ=20°).

Fig. 21 .

 21 Fig. 21 presents the relationship between the limit support pressure and the surcharge. The values

Table 1

 1 Calculation parameters.

	Cases

Covered depth C (m) Tunnel diameter D (m) Friction angle φ (°)

	Case 1~3	3, 6, 9	6	15
	Case 4~8	3	8, 10, 12, 14, 16	15
	Case 9~12	3	6	20, 25, 30, 35


Massimo De Giusti

The dancing Glossolalia of European Identity in Romeo Castellucci's Democracy in America My article aims to demonstrate that both contemporary theatre and dance offer significant means of elaborating a personal as well as collective identity in Bauman's terms of a constant process of aggregation of different aspects and trends (Bauman, 2003: 17). More precisely, I propose through the analysis of Romeo Castellucci's Democracy in America (2017) to reflect on the European identity as an epiphenomenon of today's tendencies.

Fluidity and European identity

Using a Ricoeurian terminology, we can say that contemporary identity focuses on ipséité rather than mêmeté. This means that human beings no longer recognise themselves as "in what is immutable, like [their] fingerprints" (Ricoeur, 1990: 16) i , but in an ever-changing ontological continuum. This change in the Western logical conception prompted Bauman to theorize the birth of the "society of uncertainty" (Bauman, 1999: 3) within a "liquid modernity" (Bauman, 2000: 2). According to him, "the world around us is cut into loosely coordinated fragments, while our individualised lives are fragments into a series of neverending episodes" (Bauman, 2003: 4).

Inevitably, this phenomenon has consequences in the sense of belonging to a community, more vis-à-vis with a national state. If in modernity identity was asserted mainly as a national identity, today the affiliation with a culture or a community is no longer unambiguous and immutable. As a result, membership no longer depends on origins, but is determined by a multitude of personal or external factors, which are then taken into account subjectively and ephemerally.

Upon consideration, this identity liquidity is a constant feature of European selfhood, which has, in fact, always been characterised by its instability and its provisional nature: "there has been, since men began to reflect, an immense variety of Europes" (Duroselle 1965: 68). The geographical concept of Europe has indeed evolved over time, following a heliotropic route from Asia (Homeric Hymn to Apollo), passing through Greece in The Histories of Herodotus or in The Persians of Aeschylus, to arrive at the modern doxa from Strabo (Geographika). Even the political concept of Europe has changed in both ancient and modern times, before and after the Treaty of Rome (1957) and the birth of the contemporary concept of the European Union, arguably the most important break in the conception of Europe. Only after the creation of the EEC, the borders were moved in 1973, 1981, 1986, 1990, 1995, 2004, 2007, in 2013, and they moved again in 2020 with a formal Brexit. The European identity is fluid and its spatial, temporal as well as ontological varieties have been adding and continue to add different constituent elements, depending on the eras and areas in which it is adopted (Bauman, 2008: 88; Berl, 1983: 15). As D'Andrea summarises: "the European identity turns out to be that of a civilization freed from constraints, which constantly goes beyond its internal and external limits, thus creating its own distinctive mark" (D'Andrea, 2001: 134).

, we could say that Europe has always been confronted with a crisis of (self-) consciousness and communication or relationship between its various

parts (geographic areas, cultural areas, States, citizens etc.). The European crisis seems to be a force that is at the same time centripetal and centrifugal, taking into account the chronological and semantic variations we have spoken about. One of the biggest social obstacles for the European Union project is the split between the national-states' identitieswhich are facing a similar identity crisis at the local level (Lauret, 2009: 20-31)and a European post-national identity [START_REF] Ferry | Identité postnationale et identité reconstructive[END_REF]) not yet completely conceived and felt. Thus, the European project is undermined by the "subjectivism" [START_REF] Eco | La società liquida[END_REF] of the States, caused by the lack of reference points due to the European social and identity liquidity.

Democracy in America: Language and Identity

A corollary of this identity liquidity is a deterioration of the communication and trust in contemporary (European) society (Bauman, 2004: 27). In order to propose a reflection on the mechanisms and the languages able to keep communities progressing towards unity and equality of their members, Romeo Castellucci created Democracy in America, in which he explains that: "The aim of the work is to speak about the language. This play is not political. It is not a reflection on politics as much as, if anything, its conclusion" (Iachino, 2017).

Castellucci's choice to stage De la démocratie en Amérique ii (1848) by Tocqueville is correctly dictated by the fact that in this essay the French thinker affirms that the creation of a new democratic communitymost of it being heterogeneousis inseparable from a revolution of knowledge and language, which are the real vectors of identity recognition and of internal structuring of this type of community (Tocqueville, 1848: 134, 297). It is not by chance if according to Saussure (1972: 20), language is in fact a "conventional communication system. It's a social fact, independent of the individual's actions".

Furthermore, Tocqueville (1848: 157) as well as Castellucci think that theatre "constitutes the most democratic part of […] literature" and that a linguistic and social transformation "will therefore first be discernible in the drama, and it may be foreseen that they will break out there with vehemence". According to the Italian artist, dramatic art is able to help the spectator and the community in their quest for identity because right from the beginning it has been closely linked to religion and politics: Technically, politics is born when the gods die. Politics is born when the great god Pan dies; politics is born when the party ends. Politics and all the rights that derive from it originated when we stopped dancing. At that moment, Greek theatre was born. It is a controversy over politics; it is also a way of reconsidering the function of theatre. Political models are worn out and theatre gives access to a new form of thinking that is the unthinkable. (Sebastiano, 2017: 79) From this point of view, theatre finds its origins within a revolution of the social structure of the community in which it is created. This process would push the actors as well as the spectators to rethink their personal and social identity, seek other forms of collective knowledge and use the theatre as a 'laboratory'. In Castellucci's words: When [...] the political models are exhausted, it is the theatre that gives us access to a new form of thought that is inconceivable. The theatre represents the duality of life, and does not provide models. Thanks God, there is no pedagogy. The theatre offers lights, lights in an abyss. It shows possibilities. Thanks to narration, to fiction, the theatre is able to suspend reality. This is the form of authentic liberation, of reconciliation with your body, with the bodies of others, with being together. (Sebastiano, 2017: 77) Therefore, dramatic art becomes for Castellucci a fundamental aesthetic vector of reorganisation and rationalisationin which irrationality isn't excluded a priori (Duvignaud, 1965: 88;Turner, 1982: 144)of a community facing an ontological and epistemological transformation. In order to represent and stimulate the creation of a new community, the play immerses the viewer in a time "out of language" (Pellanda, 2016), during which the languages and the community are created. Castellucci's work atomises language and identity, inviting viewers constantly to question and adapt their thoughts and perceptions, analysing every part with the same attention. It is both the personal and collective work that does not engage the audience in a precise and eternal theory of community, structure, language and identity, but stimulates them to make first-person (autonomous) decisions, to change their opinion, always to express their own ideas and feelings.

The communicational crisis of our society pushes Castellucci to reject a traditional narrative and an exclusively dialogical communication -"the modernity and its crisis marked the crisis of narration" (Lyotard, 1981: 5) in order to ask open questions without accepting any a priori or nihilistic tendency. Castellucci's modus operandi is close to Plassard's idea that "if there is a fracture, it mainly concerns the macrostructure of the text of the representation, while at the microstructural level anecdotes, tales of life, historical events, myths and other narrative topoi continue to be used" [START_REF] Plassard | Le postdramatique, c'est-à-dire l'abstraction. Theater[END_REF]. Furthermore, we could say that "instead of representing facts and acts, we find a position of sound, of words, of sentences, of noises whose logic does not derive from a single meaning, but from the scenic composition, from a visual dramaturgy not oriented purely on the text" (Lehmann, 2002: 42). It is no coincidence that the subtitle of another work, Giulio Cesare, is "pezzi staccati", which is translated as "detached pieces": it is a question of self consciousness, of order, of relationship with the others: "the order that is destroyed and transformed, the omnipresence of disorder, the creation of an organization arise from basic needs. Every theory of disorder, all theory must involve and establish the concept of organization" (Morin, 1977: 227).

Consequently, summarising this piece is quite difficult, but we can do so in this waytwo parts each made up of three scenes: (1) Chapter of the language: glossolalia; the dances with flags on each of which stands a letter; a dialogue between two Indians about the language and its role and value; (2) Behind a semitransparent curtain the actions are nuanced and indecipherable: a child in the arms of an adult who runs or moves away; choral dance; Sabbath; (3) A body without a figure, a simple stone, men (the actors are all women) must split it, build it, transform it; (4) Pilgrim peasants Elizabeth and Nathaniel measure their material and spiritual misery; the faith of the two colonists is no more than blind superstition; potatoes are rotten; she screams and is possessed by a demon; (5) A series of writings follows: of places, historical characters, biblical citations, titles, the (non-chronological) succession of the acts that make the history of the nation; (6) Two mechanical bodies twirl in the air, slow, abstract and threatening.

In Castellucci's opinion, the religious faith and the Greco-Roman heritage cannot completely answer contemporary problems in our fragmented society. Therefore, their language is no longer sufficient to create and structure a contemporary society, is no longer homogeneous, but multiplied sensu stricto et sensu lato in an eclecticism devoid of any smoothness. As in others of his works, "words are the key" [START_REF] Perrier | Portrait de Romeo Castellucci[END_REF] to Democracy in America, but they lose their uniqueness and cohesion. Castellucci exploits many forms of languages: English, Hebrew, Latin, Native North American tongues; as well as visualpaintings, tableaux, transparent curtains that form a fourth wall between the spectator and the scene, statues and light effects; acousticnoises and music; kineticdances and tableaux vivants; and finally roboticalgorithmic movements of machines. Each scene is both independent and linked to the others, never fully explained. Each form of expression is only part of a complex reflection regarding the body, objects, words, space, light, sound and chronology.

More precisely, Castellucci says that in order to search for new expressions and escape the illustrative and the trap of vitalisme, he had to rely upon dance and choreography. In a historical transitional context, what attracts the Italian artist to dance is its metamorphism evoked by Michel Bernard. As the liquidly identity, the movement is in fact never identical or fixed, but always precarious and in transformation. In Castellucci's opinion, the first specificity of the dancing body is "its dynamics of indefinite metamorphosis and the exuberance of the movement due to its own change. Dance always offers itself as a mad search for an individual body that tries in vain but constantly to deny its apparent unity and identity in the manifold, the peculiarity of its acts" (Bernard, 2001: 18). Dance is not exclusively tied to a rational organisation, but more to logic and a physics of the body and movement. It is an unpredictable logic that produces the strength and novelty of its imagination. Dance becomes a way of research; it summarises the movement that helps us understand who we really are within an increasingly unstable balance (Beauquel, 2010: 65-77). This continuous transformation introduced by dance offers an example of the instability of a contemporary society, showing the viewers its constant and irregular flux: every movement is possible, the most codified and controlled as well as the most free and irrational. We can find an example of these phenomena in the majorettes' dance in the first part of the piece, where dance acts as the mirror to a newborn society, searching for a common identity. In the scene, eighteen amateur dancers, dressed as majoretteshalf women, half soldiersshake the bells attached to their belts. Each one bears a flag or cane on which a letter is printed. Within a few moments, they first form the phrase DEMOCRACY IN AMERICA; after, alternated by short dances, they form many anagrams such as DECAY CRIME MACARONI, CAMERA DEMONIAC CRY, and finally the name of some countries democracies as well as dictatorships: CANADA MACEDONIA ROMANIA ARMENIA MYANMAR YEMEN IRAN OMAN etc. It is a precise ballet, apparently elementary, a game as much as a military exercise. The dancers form a single figure only in the first part of the dance and ephemerally, dividing continuously afterwards into always different and heterogeneous groups and subgroups. Their only traits d'union are the uniform and the freedom to join new groups.

Even the algorithmic language structuring the movement of the machines in the final scene is part of this (de)construction of language. It does not escape the arbitrary limits and therefore the asymptotic identity of any other language. As Castellucci says, the schematisation of the movements looks to symbolise a "victorious industrialization" [START_REF] Génécand | Le fabuleux sacre de la poussière[END_REF], a better future, as well as its potential dark implication: "machines feed the scene. [...] In my opinion, machines are clearly a masculine principle with the notion of order and control" (Sebastiano, 2017: 78). In other words, new technologies are not a deux ex machina, but present the same limitations of any other language. In addition, "the new technologies, creators of virtuality, lead to the rupture of every rhythmicity, whether biological or social, associated with the idea of cycle of existence, a rupture to which an idea of simultaneity, instantaneity and presentification" (Semprini, 2003: 158), which is as we have seen a feature of contemporaneity. Instead of simplifying the understanding and the sense of belonging derived from it, the new technological language does not give any definitive answer, becoming even more metaphysical and intelligible than traditional ones, in an evident reference to the technological transcendence of Durkheim (Maffesoli, 1991: 33). As Bauman (2003: 25) stated: "The virtual communities that have replaced the natural ones create only the illusion of intimacy and a fiction of community. [...] Nor are these virtual communities able to give substance to personal identity, the primary reason for seeking them". It is interesting to notice that this is the same critique that many Eurosceptic movements make against the actual European Union and sometimes against the European identity tout court (Muzzi, 2019: 7).

These negative tendencies and acts of sacrifice that underlie the playthe most important ones being the abandonment of the native language by the Indians and the abandonment of the son by the Puritan mothersummarise the sacrifice that every adhesion to a new community and to a new identity, that every creative act entails: "Starting from Cain, every act of creation carries within itself a kind of black core, a negative charge, the power of a non-being that from within undermines any claim to existence," [START_REF] Castellucci | Genesi[END_REF]. According to Castellucci, "Tocqueville was the first to study this system and to emphasize the gray areas that force us to downsize the idea that democracy is the ideal political system" (Perrier, 2016). Furthermore, these shadow areas have been the basic material of the theatre since ancient Greece: "[Greek tragedy]" says the Italian artist "is a kind of laboratory to study the shadow areas of all human existence. It is from this shaded area that the immense artistic expression of the Greek poleis was born. [...] Tocqueville thinks that one must continually pay attention and remain conscious, eyes and ears open" (Perrier, 2016). For Castellucci "there is no tragic gaze worthy of being tragic, that is, no tragic gaze capable of creating a human community with its own strength. This is the task that the theatre of the future must carry out [...]" [START_REF] Perrier | Portrait de Romeo Castellucci[END_REF].

As a result, he dynamises the spectators, who are called to adopt a personal vision, to intervene actively in the development phase itself, to the detriment of all stability and intelligibility: "The theatre is a temple and its oracles never speak distinctly. Here, the spectator works and deciphers; his unconscious too" (Sorbrier, 2017: 7). This empowerment of spectators inevitably implies that the dramaturgy "invites the spectator to reflect on his habits of perception" (Lehmann, 2002: 147) because the contemporary identity is an "effort" and a "duty" of construction "avoid[ing] the solidification and leav[ing] the options opened" (Bauman, 1995: 124).

Glossolalia

Castellucci applies this non-hierarchical exploration of dramatic possibilities that can influence contemporary social and identity relations to language through the idea of glossolalia. In the first scene a glossolalic speech, recorded in a religious congregation in Kansas in 1890, is broadcast while the spectators can read in a projected caption a passage of the Acts of the Apostles glossolalia. Here glossolalia is described as the act of speaking and praying aloud in a language having the appearance of a foreign language, unknown by the person who speaks it, or by a sequence sequel of incomprehensible syllables.

In these terms, glossolalia represents in Castellucci's play an attempt at communicating iii within a community using a language in which the meaning is given through addition, by being with others rather than by a predetermined proper sense. It is a multiple ephemeral and personal code, arbitrarily chosen iv and only partially understood, formed by words as well as images, gestures and dances. This glossolalic approach is close to a hieroglyph system, which "protects from this decline in communication because it does not compete with [normal] language. These are conceptual images, not signs that point to words. Since they don't refer to the spoken language, they cannot replace it either" (Assmann, 2018: 195). The whole show is a continuously changing and not completely understandable glossolalia, becoming a hieroglyphic practice and a rite which "as for gestures, [is] as transitory hieroglyphic, and [is] to hieroglyphics as words spoken are to words written, in that they abide not" (Bacon, [1605(Bacon, [ ] 2010: 213): 213).

Nevertheless, no structure supports glossolalic language, only inspiration, only the intelligence, sensitivity and imagination of the sender and recipient, of the artist and of the public (Castellucci, 2019). To establish these associations, the viewer must consider every stimulus, every language. As Perelli states:

Reading the dialogues outside the imaginative context in which we listen to them would be useless for the purpose of understanding because out of context they are simple, to the limit of the banal, anyone could ask themselves these same questions. This is because these same dialogues are moving images and are part of a work in which there is no preponderance of one element over the other, in which harmony is total and inseparable; [START_REF] Carraro | Democrazia in America: uno sguardo molteplice fra politica, suoni e loggione[END_REF] Castellucci thinks that a play is a: [...] contact, [a] shared creation. I deeply believe in the role of the spectator. A show is nothing, it has no object. It is the artistic language, the most fragile thing imaginable because in the end, the stage empties and nothing remains […] What remains then? The experience lived by every spectator. Each show is a sum of objects, shapes, colours, sounds, lights. But without the experience, the history, the scars of each spectator, the show is useless. And it's the spectator who brings the show to life, not the opposite; (Castellucci, 2019) We can say that the understanding of the association between 'content' and 'form' is entrusted to the viewer, the director "develops an autonomous visual dramaturgy, which loses its function of dramatic support to invite the viewer to reflect on his habit of perception" (Lehmann, 2002: 75) and, we might add, sharing a sense of recognition and belonging.

The glossolalic association is summarised by the opaque curtain that forms a fourth wall between the spectator and the scene, which performs again a hieroglyphic function of "transmit[ing] a truth by concealing it" (Assmann, 2018: 133). It is interesting to notice that traditionally this is the representation of the law and the identity in a hieroglyphic culture (Assmann, 2018: 130-134), suggesting to us why the Italian director wants to stimulate the spectators as well as the communities to reflect through a hieroglyphic, opaque representation of the "[…] dilemma of liberation from the law as the learning of freedom, to the manipulation of consciences, to the muscular strength of individualism, to the role of everyone's choices concerning the social contract which defines the rights and the rules of collective life, without these themes becoming direct discourse on democracy. The ancient Athenian model here is only a false papier-mâché frieze, and American history is a sequence of legends relating to historical events and dates with a bloody echo. [START_REF] Castellucci | Democracy in America[END_REF] The limited vision is sublimated by sounds and noises, leaving room for the imagination and the sensitivity of the spectator: "the signifying opacity not being able to make sense, makes a sign: we can only identify languages that we don't understand. Then, we can only name them, which allows a return of the meaning and breaks the suspension of the signifier" (Courtine, 1983: 56).

This personal and social rethinking is made possible by glossolalia because this type of language "[...] already implies the exteriority of a commentary, a strange feature necessary for its own empowerment" (De Certeau, 1980: 14). Consequently, glossolalia becomes a new form of "integral communication, without remainder, where the said and the meaning abolish all mediation to run through the assistance of the same dynamic from the same original impulse, the inspired discourse does not belong in its own right to any of those who utter it or receive it, who no longer belong to themselves" (Puech, 1988: 28).

In these terms, glossolaliaand Democracy in America in general -seems to be an approach in order "[...] to find an exit from uncertainty" (Bauman, 2004: 17) of personal and social life nowadays:

It's true that we compose our identity (or our own identities) as a drawing is composed starting from the pieces of a puzzle, however the problem is that the puzzle is defective and some pieces are missing. [...] No facilitation is available when we compose our identity. [...] In the case of identity, we don't start from the final image, but from a certain number of pieces that we have already come into possession of or that you think are worthy and then we try to find out how to order and reorder them to obtain a certain number of satisfactory images. (Bauman, 2004: 95)

Conclusion: A new European glossolalic krisis

As we have seen, the mobility and the alteration of contemporarypersonal as well as collectiveidentity are typical features of the conception of Europe since antiquity. We have confirmation of this phenomenon even within the current European project if we consider that the adhesion as well as the separation from EU take place as a choice and not as an imposition.

The Europeans should stop trying to crystallise something that is historically and ontologically dynamic and rather accept that it as a quintessential and innovative feature of their European identity. In order to realise what Montesquieu defined as "a nation made up of several" (Montesquieu, 2000: 88), Europeans must admit that their European identity is not formed solely from a common past, but modulated on the present and projected into the future. It is not a question of claiming this or that other historical-cultural affiliation or of imposing a certain artificial identity. All citizens and all member states are called upon for their contribution, to constantly adapt to new challenges and historical phases, to be "united in diversity" beyond any difference or identity crisis, choosing which values, symbols and memberships to belong to.

It is interesting to note that the link between European identity and the concept of scission or isolation is already present in the etymology of the word crisis, because in ancient Greek the word krisis contains the nuance of separation, differentiation and choice (Montanari, 2001: 258). Moreover, this conceptborn in the agricultural environment and subsequently applied to medical jargonhas been linked in Greek mythology with the legend of the kidnapping of Europe, daughter of a king of Asia Minor, that Zeus, transformed into a bull, takes to the island of Crete to father the first Europeans (Wissman: 2014). The very birth of Europeans would therefore be a result of a separation, of a krisis, which we still have to accept and integrate in our modus essendi et cognoscendi. The real European crisis would be the non-choice, the standardisation and the conformity.

However, the fact of choosing always implies an enormous freedom as well as a great responsibility (Baumann, 2003: 83) that we must be able to manage, to transform it successfully into an opportunity and an obstacle. Castellucci and contemporary theatre show us how not to be afraid of transformations, of crises. In order to create a true community, the Italian artist suggests we adopt an integral communication taking into consideration the various "detached pieces" to create a new voice, changeable but not necessarily chaotic. By doing so, a dynamic community is formed, in which everyonecitizens as well as national and international institutionsis called to participate directly and responsibly in decisions being made by trying to combine all the different opinions and cultures personally as objectively, logically as well as sentimentally. The risk for Indians in Democracy in America and presently for Europeans is, that the other's "words don't tell our things" [START_REF] Castellucci | Democracy in America[END_REF]. Using Max Weber's terminology, we can say that in Europe we have a Gesellschaft, but we still need a Gemeinschaft.

Notes

i All quotations in languages other than English have been translated by the author. ii Tocqueville did not want to write a simple study on the American democratic and national epiphenomenon, but rather to draw up an oeuvre de pensée operating on different temporal and thematic axes. Diachronically, the book presents the ideal form that American national and democratic community has adopted, but it also works synchronously suggesting the political line to be followed in the transition towards the birth of any new democratic organisation, as in France at the time. It is an outdated, comparative work, not contained in a single hic et nunc. iii Rousseau had already imagined a total language accepting all types of mistakes or deformations (Rousseau, 1993: 60). iv Even the theatre as a place is nowadays considered arbitrary. The semantic individualisation is associated with a reconstruction "of the spaces of convention where the human being could imagine even the most complex and ambiguous idea of reality. […] The citizen must accept the intermediate of the figure and the sign in order to participate in the political relationship. The access to the problematic community which needs the participation of the citizen involves the deviation through the arbitrary space" (Wallon, 2005: 302).