Nicolas Peyret Supmeca

Gaël Chevallier

Jean-Luc Dion

DYNAMICS OF ASSEMBLED STRUCTURES: TAKING INTO ACCOUNT THE SURFACE DEFECTS IN INTERFACES

In structural dynamics, the prediction of damping remains the biggest challenge. This paper deals with the energy losses caused by micro-slip in a nominally planar interface of a structure. This paper proposes an analytical and experimental study of flexural vibrations of a clamped-clamped beam with innovative position of the interfaces. The objective of this test bench is to characterize the global rheology of the interface. The proposed model aims to characterize this rheology based on local settings of the interface. First, the test bench is described and the choice of the position of the interface is justified. The experimental bench and the dynamic behavior of this structure are presented. We propose to illustrate the mechanism of energy losses by micro-slip by making a comparison between the behavior of a "monolithic" beam and a sectioned beam. Secondly, a modeling of the interface taking into account the surface defect is presented. The energy dissipated by friction in the interface is calculated during a loading cycle. This leads to a computation of the dissipated energy and thus to a nonlinear loss factor. Finally, we confront the loss factor calculated analytically and the measured one.

INTRODUCTION -PRESENTATION OF THE TEST BENCH

The present work starts from the idea of a previous paper [START_REF] Peyret | Micro slip Induced damping in planar contact under constant and uniform normal stress[END_REF]. Firstly the shape of the beam has been improved in order to obtain the greatest damping as possible by increasing the dimensions of the interfaces, see Peyret et al. [START_REF] Peyret | Non linear dynamic behavior modelling of a planar friction interface in a structure assembly[END_REF]. The final design of this new benchmark was presented in Dion et al. [START_REF] Dion | Improvement of the measurement techniques for the damping induced by micro sliding[END_REF]. The shape has been designed to avoid coupling between the normal stresses and the vibration motion even if the interfaces have been inaccurately located (Figure 1). Piezoelectric transducers have been glued in order to excite the first vibration mode; sensors allow to obtain the normal load in the interfaces and to measure the transverse motion. An original experimental procedure is based on stop-sine excitation. The authors justify that their excitation method is more efficient that Dirac (shock) or Heaviside (release of a static load) excitations.

Signal processing tools are proposed to post-process the instantaneous damping and frequency of the first mode of the structure. Finally this work experimentally demonstrates the presence of micro-slip damping, by comparing the vibration behavior of a monolithic part and a built-up structure with friction-joints (Figure 2). This comparison is the best way to determine the "added damping" [START_REF] Hertz | On the contact of elastic solids[END_REF].

MODELING OF THE INTERFACE: MULTIPLE SPHERICAL CONTACT

When conducting functional surfaces (geometric area) specified by the design, it appears, in manufacturing, some geometrical defects. These defects are questioning the assumption of a plane contact interfaces in a uniform normal stress distribution.

The modeling of the interface by a plane surface can not take account the fact that there are energy losses for smallest oscillations (Figure 3), this is due to the surface defect. To take into account the geometrical defects with contacts on several sphere, the contact interface is modeled as follows: contact is made with two types of sphere (Figure 4):

m spheres "G" ensuring the rigidity of the interface. They are the first to come into contact during the implementation of the normal load and can not slide completely (macro-slip) ; -n spheres "P" potentially slide completely. Interface load is defined by the normal approach δ n of the two normal planes constituting the interface. Before loading, the difference between heights of spheres is defined by h. The spheres are defined by their radius of curvature ρ G et ρ P .

Position of the contact plane

After normal loading of the interface Before normal loading of the interface Spheres G and P, before and after the loading of the interface normal The distribution of forces on the interface is defined by: normal load N and tangential load T. The distribution of these forces on different spheres constituting the contact by N i and T i .

δ n h ρ G ρ p N G N T N P T G T P

Forces distribution on the spheres

G P G P N mN nN T mT nT = + = + (1)
The distribution of the normal load is a function of the normal approach δ n and is defined, for one spherical contact, by the Hertz theory [START_REF] Hertz | On the contact of elastic solids[END_REF]:

2/3 * . 1 3 . 4 i n N E ρ δ ρ       =        (2) With 2 2 1 2 * 1 2 1 1 1 E E E υ υ - - = + 1 2 1 1 1 R R ρ = + (3)
Using the forces distribution given on figure 4, the normal load on the interface can be expressed by:

( )

1 3 1 3 2 2 2 2
4 * 3

G n P n N E m n h ρ δ ρ δ       = + -        (4)
Once the normal load is applied, a tangential displacement δ is applied between both surfaces of the interface. The assumption is that this displacement is the same for each contact. The Mindlin theory [START_REF] Mindlin | Compliance of elastic bodies in contact[END_REF][START_REF] Mindlin | Effects of an oscillating tangential force on the contact surfaces of elastic spheres[END_REF], gives the displacement from the partial slip into the contact surface (defined by c on Figure 5):

( )

2 2 * 3 3 16 i a c N G a µ δ - = (5) with * 1 2 1 2 2 2 G G G υ υ - - = + and
( )

2 1 j j j E G υ = + (6)
From the previous equations, Johnson [START_REF] Johnson | Contact Mechanics[END_REF] offers solutions for a spherical contact, initially loaded with a constant normal load Ni, and subjected to a tangential load Ti oscillating between ±T max with max i T N µ < (Figure 6). The surface defined by the cycle gives the dissipated energy for a cycle. This energy can be expressed as a function of the tangential displacement δ:

i i i i diss i i i i N a a a W G a N G N G N G µ δ δ δ µ µ µ                                   = -- - -- -                                            (7)
From the Mindlin theory [6-7] δ ilim is calculated for one tangential load. δ ilim define a limit value of tangential displacement for each type of spherical contact: passage from macro-slip phase to total slip phase.

* lim 3 16 i i i N G a µ δ = (9)
This limit is defined on the loading cycle with macro-slip (Figure 7). In function of the value of δ, two phases can be highlighted:

-Spheres G and P did not slip totally: phase called "generalized partial slip" (GPS); -Spheres G remain in partial slip and spheres P slip totally: phase called "pseudo-partial slip" (PPS).

The distribution of the tangential load, is function of the normal approach δ n :

For the phase of generalized-partial slip (GPS)

3 3 2 2 * * 16 16 1 1 1 1 3 3 G P G P G P a a T m N n N N G N G µ δ µ δ µ µ                         = -- + --                                 (9)
For the phase of pseudo-partial slip (PPS)

3 2 * 16 1 1 3 G G P G a T m N n N N G µ δ µ µ             = -- +                 (10)
The loading cycle of the multiple spherical contacts are plotted on figure 8. Comparing the shape of the cycle to that given by a single spherical contact (Figure 7) shows a new behavior during the loading especially when the spheres "P" begin to slip completely.

CHARACTERIZATION OF THE JOINT SURFACES OF THE TEST BENCH

To obtain the real contact surfaces of the interfaces some 3D measurements of these surfaces were made.

On each surface, a million points were measured. After filtering and re-sampling, we obtain a mesh of points on the surface of 1200x800 dots.

The contact being made on the more external spheres of the material, and so as to define a cloud of points defining possible areas of contact, it was chosen to retain only the "higher" points (10% of the set of points).

Figure 9 shows that the surfaces are composed of several spherical zones on which contact will be established. To determine the influence of defects of forms on the damping, the interface is modeled. We model the spheres of the surfaces as follows:

df R ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅/2 ∅ ∅ ∅ ∅ df

Figure 10. Geometric modeling of the sphere (left) -Identification of the parameters of the sphere on the measurement (right).

With R: radius of curvature of the sphere, df: height of the sphere and ∅ the diameter of the apparent surface of the sphere.

The radius R is then calculated using the following expression:

2 2 4 2 df R df   ∅    +         = (11)
Two types of sphere are identified in accordance with the model: These two types of spheres exist on each surface. During the normal loading of the interface and considering the position of each sphere, it is considered that the spheres come in contact with a surface considered flat. The spheres G are the first and most compressed, they ensure the rigidity of the interface. The spheres P are slightly compressed.

Spheres Number df (m) ∅ (m) R(m) G n = 5

LOSS FACTOR

The equation defining the energy dissipated by one spherical contact ( 9) and ( 10), are used to define the energy dissipated by the multiple spherical contact:

For the phase of generalized partial slip (GPS)

( ) ( ) ( ) ( ) ( ) ( ) 2 5 3 2 * 2 2 2 5 3 2 2 * 9 5 1 1 1 1 2 ... 10 6 5 ... 1 1 1 1 2 6 16 3 G diss G G G G P P P P P i i i N W G m A A A a N n A A A a a avec A N G µ δ δ δ δ δ δ µ               = -- - -- -                                 + -- - -- -                 = (12)
For the phase of pseudo-partial slip (PPS)

( ) ( ) ( ) } 2 5 3 2 * 2 2 * 9 5 1 1 1 1 2 10 6 16 3 G diss G G G P G i i i N W G m A A A n N a a avec A N G µ δ δ δ µ δ µ               = -- - -- - +                   = (13)
The energy dissipated is plotted on figure 11. Taking into account the external force applied on the structure to obtain the dynamic behavior, it is possible to define the loss factor of the structure:

2 diss ext W W η π = (14)
The energy dissipated by the load cycle W diss is defined by sections (equations ( 12) and ( 13)) depending on the value of the tangential displacement δ and δ ilim (equation ( 8)) The work of external forces W ext can be expressed in terms of displacement at the center of the beam υ:

2 1 . 2 ext m W K υ = (15)
with K m = 2.81 10 7 N/m : the modal stiffness of the structure. This gives the work of external forces applied to the structure:

2 max 1 . 2 ext m mcis T W K K       =        (16)
with K mcis = 3.23 10 7 N/m : the shear modal stiffness of the structure.

We can plot the loss factor evolution of the structure for a normal pre-load of 460 N on the interface (Figure 12). The transition from GPS to PPS leads a drastic change in the evolution of the loss factor. To compare the experimental results with the results provided by the model, we need to isolate the contribution of interfaces studied of the contribution of the other sources of damping. For this part the experimental results were linearized by parts. The curves in Figure 13 show the evolution of the damping of the monolithic structure (ref. beam) and of the assembled structure in the case of a normal load of 1000 N and 200 N. The two structures, assembly and monolithic, have the same geometry and the same boundary conditions, it is thus possible to consider two distinct sources of damping. We distinguish the dissipation induced by the interfaces f d2 of the dissipation provided by all the others sources f d1 (clamps, intrinsic damping of the material ...):

1 2 ( , ) ( , ) d d mx kx f x x f x x + = + ɺɺ ɺ ɺ (17)
The dissipated energy can be written as:

( ) 1 2 ( , ) ( , )
diss d d W f x x f x x dx = + ∫ ɺ ɺ (18)
Both energies are separable by subtraction. It is considered that W d1 can be identified by a test on the monolithic beam and that W d2 can be calculated by subtracting the damping rate of the monolithic beam to those of the beam assembled. Thus the contribution of the two interfaces is isolated in the damping of the structure (Figure 14). We can compare these experimental results with those given by the model (Figure 15). We note that we reached the same levels of damping, although this level is reached experimentally for more smallest oscillations. This difference is probably due to the choice of a contact model at two types of spheres of different heights. Indeed this model does not take into account the progressivity of the spheres height in the interfaces and the fact that probably some irregularity of the surface outcrop the contact and dissipate some energy from smallest amplitudes of vibration.

CONCLUSION

Unlike a modeling that takes into account the interface as a plan geometrically perfect [START_REF] Peyret | Micro slip Induced damping in planar contact under constant and uniform normal stress[END_REF], taking into account the surface defects it is possible to model the presence of damping in the interfaces as soon the smaller amplitudes of vibration.

A model that takes into account surface defects was developed to more accurately model the evolution of the damping as a function of the amplitude of the deformations. This new approach is a multi-spherical modeling of the interfaces, based on the behavior of the spherical contact developed by Hertz and Mindlin. This modeling allows us to compare the loss factor found experimentally

1

 1

Figure 1 .

 1 Figure 1. View of the benchmark screwed on its Steel Surface Plate. Part 1 is a massive part designed to clamp the beam on the heavy Steel Surface Plate. Parts 2 are the strain gauges glued on both sides of the beam to measure the normal load. Part 3 are the piezoelectric patches glued on both sides of the beam to excite the first vibration mode. Part 4 is a quick clamp used to apply the normal load before the tests. Part 5 is the Bruel & Kjaer 4517 miniature accelerometer.

Figure 2 .

 2 Figure 2. Evolutions of damping versus the normal load and versus the displacement magnitude. Both the monolithic (ref. beam) and the cut beam were tested. The latter was tested under four normal load levels in the range 100-1000 N

ηFigure 3 .

 3 Figure 3. Loss factor η (%) vs υ (m) displacement at the center of the structure for plane surface.

Figure 4 .

 4 Figure 4. Modeling of the interface.

Figure 5 .

 5 Figure 5. Schematic elastic spheres pressed by a constant normal force N i and subjected to an oscillating tangential force T i .

Figure 6 .

 6 Figure 6. Loading cycle for one spherical contact. Tangential load Ti (N) vs δ displacement in the contact (mm).

Figure 7 .

 7 Figure 7. Loading cycle with macro-slip for one spherical contact. Tangential load Ti (N) vs δ displacement in the contact (mm).

Figure 8 .

 8 Figure 8. Tangential load Ti (N) vs diplacement δ (m)

Figure 9 .

 9 Figure 9. Measures three-dimensional surfaces of a testbench interfaces (measurements in mm).

Figure 11 .

 11 Figure 11. Energy dissipated W diss (J) vs diplacement δ (m)

Figure 12 .

 12 Figure 12. Loss factor η (%) vs υ (m) diplacement at the center of the structure for one multiple spherical contact surface.

Figure 13 .

 13 Figure 13. Linearized experimental damping ξ (%) vs υ (m).

Figure 14 .

 14 Figure 14. Loss factor η (%) vs υ (m) diplacement at the center of the structure for multiple spherical contact surface.

Figure 15 .

 15 Figure 15. Loss factor η (%) vs υ (m) diplacement at the center of the structure, comparaison of experimental and model results.

Table 1 :

 1 Model parameters

			1,70E-06	1,80E-02	2,38E+01
	P	m = 18	1,20E-06	1,60E-02	2,67E+01
			h = 2.4E-07 m


Goeller Adrien
email: adrien.goeller@supmeca.fr

Jean-Luc Dion

Ronan Le Breton

Thierry Soriano

Bernard Roux

Adrien Goeller

Jean Luc

Dion -Ronan Le Breton

Vannier-Kinoptik Antony

Dynamic system stochastic identification mixed with video processing: Validation on a real case

Keywords: Dynamic systems, stochastic identification, computer vision, Kalman filters

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

I. INTRODUCTION

In several parts in the world, manufacturers are expected to make a revolution. With the explosion of the Internet of Things on the consumer market, and the improvement on domains such as augmented reality, virtual reality, or robotics, manufacturing is being redesigned in several development smart industry programs. Leaders companies are already taking the leap and investing in new methods of manufacturing. Improvements of costs, better quality monitoring, new diagnostic tools, data management, the field for innovation in those industries is very large. Cameras are particularly one of the smart sensor of those manufacture method by improving quality, traceability or manufacturing time. In this way, the development of a system able to mix complex mechanical model directly with frame from camera could open the path for the machine to understand the filmed scene and process augmented variables as prediction, interpolation, or estimation of hidden variables.

Mixing data with models is not a new trend. Marchand & al. [START_REF] Marchand | Recalage en temps réel de modèle mécanique par couplage entre filtre de Kalman, erreur en relation de comportement modifiée et réduction de modèle[END_REF] use UKF to do identification and model calibration, Naets and al. [2] use accelerometers to display an augmented reality measurement on a vibrating beam, Simon [START_REF] Simon | Aircraft Turbofan Engine Health Estimation Using Constrained Kalman Filtering[END_REF] applied it on complex model. Kalman filters have been used in image tracking [START_REF] Xu | Robust object tracking using Kalman filters with dynamic covariance[END_REF] but with linear models, but this is Particle filter which is most used in video processing to calibrate models (see [START_REF] Choi | 3D textureless object detection and tracking: An edge-based approach[END_REF]).

Following the development initiated in a previous publication [START_REF] Goeller | Real Time Dynamic System Stochastic Identification in Video Capture for Data Compression, Image Interpolation, Prediction, and Augmented Reality[END_REF], this article carry on the promising concept to enable vision machine dealing with all the points mentioned before by improving the comprehension of the object by the computer. Based on a physical model of the object seen, this algorithm enables the machine to understand the behavior of the scene and to extract some hidden physical parameters directly from observation. This article introduce those concept are applied on a case study of a ruler sliding on a table.

After a general overview of the algorithm, the model of the case study is described. Kalman filters and extraction process are developed after. Then the experimental setup is explained before the results.

II. CASE STUDY

A. Algorithm overview

The algorithm introduced here is composed of 3 main parts described on Fig. 1. The important part is that every block is designed to be real time with architecture frame to frame. In spite of the current results which are not yet real time, the aim is to switch to real time during a future code optimizing phase. The video is respecting a classical frame of object detection. We took a static background hypothesis. It allows using background subtraction technic [START_REF] Yilmaz | Object tracking: A survey[END_REF]. Here the detector used is a GMM (Gaussian Mixed Mixture) [START_REF] Stauffer | Adaptive background mixture models for real-time tracking[END_REF] and gives a binary mask of the ruler. Extracted information is the center position of the object and the main orientation angle α using the statistical moments [START_REF] Hu | Visual pattern recognition by moment invariants[END_REF]. From those position are calculated the object center speed and angular speed. Length and width are measured directly on the image and correspond to the variables. The state vector used later in the assimilation part is created from those extracted values.

The assimilation process is composed by two parts. First, a decision process is designed to initialize and manage the second part. This first part is based on the number on consecutive frames where the object is detected. The second part is an Unscented Kalman Filter (UKF) [START_REF] Julier | Unscented Filtering and Nonlinear Estimation[END_REF]. This architecture of Kalman filter management allows to run the Kalman filter without observation and to be more robust.

Finally, the video output is augmented with the data from the dynamic model using a 2D model of the ruler. Here, the 2D model chose here is a simple rectangle. In order to separate the detection errors and the filter errors, two rectangles are displayed, one for the detected object, one for the assessed object ruled by the filter result (see Fig. 2) Fig. 2: Augmented video with real ruler in black, extracted ruler in blue, assessed ruler in red and length assessment in green.

B. Model description

The model chosen to simulate the behavior of the ruler is a double sliding contact point, A and B in the Fig. 3: ruler consideration for study, which models the ruler resistance by solid friction. TA and TB are the friction forces applied on A and B.  The table surface is considered to be perfect which imply that the friction coefficient is considered constant.  The 2 contact points are symmetric regarding the ruler gravity center.

III. EXTRACTION PROCESS

A. Gaussian Mixture Models (GMM)

The GMM has been first proposed by Stauffer and Grimson [START_REF] Stauffer | Adaptive background mixture models for real-time tracking[END_REF]. It is first a method to model an unknown distribution with a sum of weighted Gaussian. Let's explain with an application related to object extraction. We define for a particular pixel (u0,v0) an history of intensity grayscale value: I = {I1,…,It}. 3. Use the connected component in order to consolidate the detection. 4. Update the GMM with the "Background" labeled pixels.

The output of the GMM is a labelled frame, which means a frame which groups the different binary masks of the objects detected.

The GMM detection is very efficient for detecting moving object which appears to be as transitory signal regarding the background model. There is a limit when the object becomes static: the object is progressively integrated in the background. To become more robust, the solution is to provide more frame in the initialization phase. Some authors add a tracking step [START_REF] Stauffer | Learning patterns of activity using real-time tracking[END_REF] or a shadow detection step [START_REF] Kaewtrakulpong | An improved Background Mixture Model for Real-time Tracking with shadow Detection[END_REF] to consolidate the GMM detector.

B. Statistical moments method

Statistical moments are here used only to find basic mechanical parameters of each object: centroid, area, and orientation. Only few parts of the Hu moment are described here, regarding the needs we had.

Considering a 2D continuous function f(x,y), the raw moment is defined as :

= ( , )
Thus the discrete equivalent to apply in a numeric image is:

= ( , )
I is the 2 variables function describing the intensity of each gray-level pixel.

In this framework, some moments refers to interesting images properties for a given object in the image: M00 as the area, or the centroid ( , ) = ( , ).

To obtain the orientation, we place in the object reference by defining the central moments M'pq as for the discrete version:

′ = ( -̅ ) ( -) ( , )
A covariance matrix of I(u,v) can be construct from those moments. Defining:

= ′ ′
The covariance matrix: ( , ) =

From this matrix, the major axe of the object is given by the eigenvector of the largest eigenvalue, which means for the object orientation:

= 1 2 arctan 2 -

IV. STOCHASTIC IDENTIFICATION

A. Kalman Filters Kalman filtering refers to a family of algorithms that tracks the temporal evolution of a dynamic model based on noised measurements. This evolution is described here in the discrete time domain:

= ( , ) = ℎ( , ′ )
The first equation of the system above is the propagation equation whereas the second is the observation equation. Xk is the state vector at the step k, f(.) is the model function, wk the process noise assumed to be Gaussian N(0,Q), and Zk is the observation vector, h(.) the observation function and w'k the observation noise assumed to be Gaussian N(0,R).

The Kalman filter is an efficient algorithm able to predict the future state vector using its means and covariances. It is based on a model of the system described by the function f(.) compared to the observation Zk with the observation function h(.). The Kalman filter is based on two major hypotheses: the model function and the observation function are linear, and the variables considered in the system are Gaussian. Due to the linear hypotheses, f(.) is the F matrix and h(.) the H matrix in the following.

An approach of Kalman algorithm is to separate it in two phases. The predict phase gives an a priori estimate of the state and covariance based on previous time step tk:

B. Unscented Kalman Filters

Kalman has been derived in a lot of different forms in order to pass through the problem of non-linear cases. Among those methods, the Unscented Kalman filter method (called UKF) has been chosen to deal with the two issues of our case: the mechanical model is highly nonlinear, and it is described by unsolved differential equations.

Unscented Kalman filter has a totally different approach compared to others filters as Extended Kalman filters [2]. The Extended version proposed to linearize using a first order development of the nonlinear function. The Unscented version proposes to use the probability distribution to approximate the nonlinear function.

The unscented transform enables the propagation of the state vector through the model function f(.) or the observation function h(.). In this way, a deterministic sampling technique is used to catch the state probability distribution and the two first moments. The sampling set is composed of sigma points and those sigma points are propagated through the nonlinear function. The minimal number of points is n+1 with n the size of the state vector. In our case, it has been chosen 2n+1 sigma points. After that, the inverse unscented transformation is used to obtain the state vector and the covariance from the sigma points. This method can be repeated for the innovation phase whether the observation function is nonlinear too.

After the unscented step, the standard Kalman filter can be applied. Another unscented step is added around the innovation phase whether the h(.) function is nonlinear.

V. EXPERIMENTAL SETUP

A. State system choice

The state equation transform is a crucial step in the model process. Regarding to the model equation and description (Fig. 3 and Fig. 4), we can separate the state vector in 3 different parts:

 The observed variable from the camera: object centroid ( , ), object angle ( ), and ruler length ( ).  The model is using the translation and rotation speed as intermediate variables. These speeds have been added into the state vector in order to have a smoother value: object center speed and angular speed ( , , ).  The equation parameters: the friction coefficient μ and the position of the virtual points L1 and L2.

The motion equations are quickly complicating to a nonlinear form due to the intervention of sinus/cosinus for the intervention of α. Moreover the differential system is a coupled one, making an explicit solution impossible. All these points lead to associate the Unscented Kalman filter with a Runge-Kutta 4 solving method. The adopted approach is to make an observation matrix simple and linear and transfer all the nonlinearities to the model function.

The state vector and the observation vector at discrete time k are:

= ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ and =
The propagation matrix is linear and has the form:

H =
1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

B. Experimental setup

The test bench used to film the ruler movement is described on Fig. 5 below. The setup of the experimental was defined in order to respect some hypotheses to simplify the video processing and to respect the framework used on the simulated ruler:

 A static background  The real scene projected on the camera sensor can be approximated to a 2D scene.  The object is considered to have the same aspect during the scene which means no reflection

The sensor plane has been set parallel to the table and the thickness of the ruler is about ½ mm: this setup is avoiding shadows and perspective changes.

In order to avoid reflection, the light was set up with an angle of 30° in order to light up the table and keep the ruler as a dark object.

In order to solicit the whole model, 3 different scenarios have been chosen:

-Quasi exclusive translation movement -Quasi exclusive rotation movement -Mixed translation/rotation movement VI. RESULTS

A. Extraction

The method used to quantify the results is to compare the area detected by the GMM detector and the real area measured on the frame.

The measure consists on extracting a first binary image from the detected polygon and a second binary image from the frame threshold with an adapted value to extract only the ruler. An "exclusive or" function allows to identify the pixels P which belongs to the not overlapped area by the two extracted masks (in white on Fig. 6). Then, the matching ratio is:

ℎ = ∑( ∈ ) ∑( ∈ )
Fig. 6: "Exclusive or" between the real ruler in orange and the detected ruler in blue

As we can see on Fig. 7 and Fig. 8 the extraction has good performance. There is one non-detection because of a parasite reflection, and the detection is not complete when the ruler appears in the frame and when it begins to disappear because of its static behavior.

B. Assimilation

The assimilation part can be characterized in the restricted framework of our model of two kind of comparison.

1) Tracked variables

The variables observed. In our case it is the position, angle and ruler length. In the table below (Fig. 9) are summarizing the performances using the same method than the extraction (see Fig. 10).

2) Assessed variables

The variable linked to the observed variables by the model. It is the centroid velocity and the angular velocity. In order to do a comparison, the different velocities are also computed directly from the video processing data.

We can see on Fig. 12 that the assessment of the angular velocity is efficient and robust compare to the video processing computed angular velocity.

C. Estimation

The model has 2 parameters: the sliding friction coefficient and the 2 symmetric contact points. The sliding friction coefficient has been experimentally determined at about 0.88. We can see on the Fig. 13 that the value estimated is converging to the expected value which is the experimentally determined. VII. CONCLUSION This article applies a concept of mixing dynamic models with video processing. The concept is developed here in 2D and for known objects with a known scenario in the filmed scene. The aim is to enable the camera to understand the mechanical behavior of the object in order to do prediction, interpolation, compression and augmented reality. The results developed here show the feasibility of the concept applied on a real case study of a ruler sliding on a table: variables tracking, variable assessment and parameter estimation.

Fig. 1 :

 1 Fig. 1: Macro architecture of the algorithm

Fig. 3 :Fig. 4 :

 34 Fig. 3: Ruler parametrization for the study

  The aim here is to assess which intensities are most probably in the background. Supposing we want to model a group of pixel by K Gaussian, let's define a canonical basis of dimension K based on those Gaussian : {e1,…,eK} We define a hidden random variable ∈ { , … , } and the probability P(Z = ek) = βk. So when Z = ek, X is projected onto the k th Gaussian : I ~ N(mk,Σk) with mk the mean and Σk the covariance of the Gaussian. Which gave for ( ) = ∑ ( | ). ( ) = ( = ). ( | = ) = . ( | , Σ ) This means that the probability of X can be formulated as a sum of K weighed Gaussian. The problem consists on optimizing the values (βk,µk,Σk) for each ∈ {1, … , } The procedure of initialization is the following: 1. Define the N first frame without the object to find a set of the values (βk,µk,Σk), ∈ {1 , … , }. In our case N = 5. 2. After those N first frame, for each frame, classify each pixel into the category: {Foreground, Background}.

  The update phase corrects the deviation of these estimations based on new observations at time stem tk+1. It is composed by an innovation phase:

Fig. 5 :

 5 Fig. 5: Test bench

Fig. 7 :

 7 Fig. 7: Performance of detection during the movement without the nondetection

Fig. 9 :

 9 Fig. 9: Performance of assimilation compared to detection during the movement without the non-detection

Fig. 11 :

 11 Fig. 11: Comparison between the assimilated position and the real position

Fig. 12 :

 12 Fig. 12: Comparison between the assessed angular velocity and the angular velocity computed from detected angles

Fig. 13 :

 13 Fig. 13 : Comparison between the estimation value of the sliding friction coefficient and the expected value