Research Of Wear Measurement Of Skate Edge Based On Neural Network

1. INTRODUCTION In skating motion, in order to improve the efficiency and quality of skate sharpening, it's necessary to develop automatic skate mullers whose key technique is how to auto-detect the edge wear. Paper[1] put forward a method that approximated the edge to an ellipse, and got the theoretical formula between the scattered light intensity and the radii of the edge. With the result, applying both CCD non-contact measurement and the theories of neural network, a method for measuring the edge wear of skate is put forward in the paper. Due to the complicated nonlinear relationship between the edge radii and the scattered light intensity, it is hard to express the relation by usual regression analysis. Since the neural network can resolve non-linear problem effectively and need no accurate mathematical expression, it is adopted to realize the mapping from the scattered light intensity to the radii. If the network has been trained by a suitable training method, inputting the scattered light intensity to the network, the radii can be obtained directly from the output layer, then the wear can be worked out. 2. PRINCIPLE OF WEAR MEASUREMENT In skating, the skate edge will be abraded in different degree because of different truck and different kinds of skates. With approximating the edge to an ellipse, there is a theoretical relationship between the wear and the scattered light intensity, by which the wear can be obtained if the scattered light intensity is given. 2.1 Theoretical measurement formulas Shown in Figure 1, the ellipse model of the skate edge is set up in rectangular axis, where a ,b respectively presents major semi-axis and minor semi-axis of the ellipse; x ? z plane is the cross-section of the edge; y is the direction along the edge; B is the tangent point of profile or working side with the edge; αis the internal angle of OB with z -axis, where O is the origin; h is the edge wear , and the internal angle of profile with working side is 90°. The cross-section shape of the edge is described by mathematic formulas as below: ??? zx == abcsionsθθ| θ|≤α| y |≤H2 (1) where, H is the edge length investigated; and arctan( 2)2aα=b (2) In Figure.2, the scattered light from the edge forms the diffraction pattern in reflex space. It is theoretically inferable that the scattered light intensity is related to a ,b as follows[1]: 20222202022' 0(sinsin)]|()|2I ( p)= λUrsbHsinc[2λπH?+??Ψ(3) where, P ' ( x ' , y ' , z ')-an arbitrary point in the reflex space; U 0 -the amplitude per unit length of light source; λ-the wavelength of the incident light; r0 -the distance from the origin of coordinates to light source; s0 -the distance from the working profilehB B a b O xzyαthe edge Fig1. The ellipse model of skate edgeQ ΨP( x,y,z)P' ' (x',y',z') αythe edge zxO Fig.2 The sketch of diffractionorigin to P '; c -the velocity of light; ? -the internal angle of the scattered light with x ? z plane; ? 0 -the internal angle of the incident light and x ? z plane; Ψ-the internal angle of the scattered light and y ? z plane, and ∫+??Ψ=Ψ++Ψαα( )R (θ0 )exp{ik[bsinsinθ0a(cos?0cos)cosθ0]}cosθ0dθ0 (4) where, θ0 is the internal angle of the line OQ and y ? z plane, in which Q ( x0 , y0 , z 0)is a point on the edge; R (θ0)is the reflectivity of point Q ( x0 , y0 , z0 ); k =2π/λ. Because the angle of working side with profile is 90°, the wear is obtained from formula 1 and 2 as follows: 2222abhb= + (5) Then the relationship between the wear and the scattered light intensity is known. If the scattered light intensity is given, the edge wear can be computed indirectly according to formula 3,4 and 5. 2.2 Method for measuring the scattered light intensity The linear CCD is applied to measure the scattered light intensity, and its validity is proved as follows. The basic unit composing CCD is MOS called photosensitive unit whose output electric charge is expressed as QI P =ηq?neoATC (6) where, ηis the quantum efficiency of materials of the photosensitive unit; q is electron charge; ?n eois the photon beam velocity of the incident light; A is the area of the photosensitive unit; Tc is the immitting time of the light. When CCD isselected, η,q and A are all constants ,and CCD drivers make the impulse-transfer period invariable, that is, TC is also constant, then the output electric charge QI P is proportional to the photon beam velocity ?n eo. It is reasonable to suppose that the injected light intensity on the single photosensitive unit is a constant, letting the incident light is homochromous, then there is a linear relationship between the photon beam velocity and the light intensity, such that ∫?? =Φ=d?vIne o 2πh eλv 2πehλ(7) where h ,νand λare constants. With above formula, (6) can be rewritten as follows Q IP = ηqAπTcvIeλ∫d?=K?Ieλ2 h? (8) where K is a constant. On the other hand, the output voltage of the CCD is proportional to QI P, so U = Ke ?Ieλ,where K e is a constant. Therefore, the linear CCD can be used to measure the scattered light intensity. The measurement sketch is shown in Figure 3,where, L is the distance from the CCD to the edge; P ' is the receiving point on the CCD the reflex light; l is the distance from P ' to the central axis . lprofile working side P'Ψincident light reflex light central axis L CCD Fig.3 The sketch of measuring the scattered light intensity by CCDAfter normalization, the relative scattered light intensity is expressed as: 22000002exp{[sinsin(coscos)cos]}cos|(0)|| ()|∫+?I r = ??Ψ=KrααikbΨθ+a?+Ψθθdθ(9) where, 22000020 1/exp[(cos1)cos]cos|(0)|| ()|∫+?=?=+ααKr Rθika?θθdθ. Because the range of variable Ψis confirmed, the vector I r is only determined by ( a , b ), and the value of ( a , b) can be obtained from I r. 2.3 Computation of arguments a ,b with neural network Formula 3 shows a complicated non-linear relationship between the scattered light intensity and arguments a ,b . The arguments cannot be directly obtained through simply processing the scattered light intensity got by CCD. Therefore, the neural network is applied to compute a ,b , which has the capability of approximating an arbitrary continuous function if the transitive functions of its neurons are continuous. Shown in formula 9, the one set of ( a ,b) determines one and only I r vector. On the other hand, if a I r vector is given, ( a ,b ) can uniquely be obtained from I r. The number of training samples will increase exponentially along with the input dimension of the network. Due to lots of CCD photosensitive units, I r vector has so high dimension that it cannot be directly inputted to the network. Therefore, K-L transformation is here used to extract the features of I r vector. After replacing I rwith its features, there is a mean square error that is sum of small discarded eigenvalues of the covariance matrix of I r samples, that is ∑=+= Dimi1ε2 λ(10) where, D is the dimension of I r; m is the dimension of extracted features; λi , i = 1,...,D are eigenvalues of the covariance matrix. Table 1 The former 9 eigenvalues of the covariance matrix(sorted from large to small ) 0.90279 0.40944 0.24349 0.042902 0.00746 0.00129 0.00022356 6.3556e-005 2.3629e-005 …For example, table 1 shows a set of eigenvalues of acquired I r samples. Because the sum of eigenvalues after 4th is 0.0091, if ε2 <0.01 is wanted, the features dimension is only 4, i.e. I r is compressed to 4-dimensional vector marked by p . Then, a non-linear mapping from p to ( a ,b) is set up as follows: ( a ,b)= f(p) (11) where, f (?) is a non-linear function that is directly related to formula 9. Above mapping can be approximated uniformly by a hidden-layer feedforword neural network whose input is the vector of eigenvalue coefficient p of I r and output two-dimensional vector ( a ,b ). The sketch of the neural network is shown in Figure.4. If ( a ,b ) is obtained by above network, the wear h of the edge can be computed from formula 5.