where C wp w p , i ,1 ,1 i , i [ ,, ]iiin i n T is the vector composed of the control points of the i-level NURBS curve. According to the principle of wavelet decomposition, φi can be decomposed into φi-1 pressed as: and ψi-1 ii 11i PQ is the nodal interpolation matrix, and it can be calculated by Oslo method [11]. connects the function in the space Vi where P nn ii 1 Q nnni () 1ii function in Wi-1 by Wang [12]. expressed as: 1, ,n k nn i 11i , 0 , j i1 , 1, , ii1 (5) Therefore, the i-level NURBS curve can be decomposed into the sum of i-1 level NURBS curve and B-spline wavelet curve belonging to space i−1. It can be expressed as: r u uC u P uQ Ci i() () ( () () ) T T TT T i iii 11 (6) For some complex surfaces, the distance and arc length of sampling points are directly related to the characterization of the surface complexity. The curve is parameterized, and the complexity of sampling points is comprehensively evaluated by calculating the arc length and B-wavelet parameters of the curve. The calculation formula of parameter curve in 11 t tt , n is: l t r u du( ) () t 1 t (7) The approximation difference between continuous curves can be measured by the value of B-wavelet curve. In this paper, we use the square of B-wavelet basis function in [9] to measure the complexity of shape. Since this value is a function on B-spline curve r(t), Ei is a matrix that with the B-spline wavelet , which can be solved by the method proposed Using the orthogonal relation between φi-1 and ψi-1 , Q can be (4) . This process can be exwhere l(tn+1 of l(t) and () definition curve based on the mixture of these two quantities is: c t w wE t lt () 2 () lt () 1 le n 1 () (10) ) is the sum of arc length of curve, and the values Et are coded in the range of (0,1) by normalization algorithm. wl and we are the weights of arc length and energy function, and the value in this paper is 1. For the sampling process of a complex curve, the spacing and density of initial sampling points can be rezoned by fixing the value of c(t). The complexity of sampling points is positively correlated with () curve is large, the value of () Et . When the local complexity of the Et increases. In (10), the sampling spacing is adjusted by fixing the coefficient on the left side of the equation. When the value of l(t) decreases, the sampling spacing becomes smaller. On the contrary, the distance between sampling points becomes larger. Experiment and Results (t) is defined as the cumulative integral of arc length and shape measure, which represents the energy of B-wavelet basis function along the curve. The energy formula of the curve is: nn E t ijt j () ( ) ( ) t ii 1 1 2 t r u du 1 The mean value method is used to express the numerical relationship between the local energy of the curve and the overall energy of the curve. The energy mean function is: n Et ET nE t () i 1 i() in1 () (9) where Ei(tn+1) is the sum of all the values of the ith B wavelet curve. In conclusion, based on [9], the cumulative arc length and the function based on B-wavelet can be calculated through (7)-(9). The complexity relationship of the parameterized August 2022 (8) Based on the section line method, data sampling experiments are carried out on the spherical crowns. The spherical crowns usually have the characteristics of semi ellipsoid shape. According to the structural characteristics of the end surface of the parts to be tested, the parabolic curve is used to approximate the longitudinal section of the spherical crown. Based on the abscissa of the parabola, the radius of the measurement track circle in the sampling plane is planned. The workpiece to be measured is an equal scaled sample of a large ellipsoidal workpiece in aerospace fields, and the diameter of the workpiece is 400 mm. The sampling strategy proposed in this paper is used to plan the distribution of sampling points on the parabola. The non-contact measurement system is adopted for the data acquisition experiment of the workpiece to be measured. The profile measurement system mainly includes the serial manipulator, measurement sensor, sensor installation device and high-speed analog input module (DMP) of a CNC system, which is mainly used to complete the data measurement of each profile. The profile measurement system, the workpiece and the sampling results are shown in Fig. 2. The non-contact laser displacement sensor is fixed at the end of the manipulator and used to measure the profile data. The position of the laser sensor can be changed by controlling the manipulator, and the whole surface of the workpiece can be measured by planning the measurement trajectory. The measurement trajectory is planned in the reference plane by using the sampling point planning results in Fig. 2, and the data simulation is carried out by MATLAB 2018b. The measuring track of the measuring device consists of k concentric circles. According to the results in Fig. 2, the number of concentric circles is set to 20. Taking the uniform sampling method as the control group, the number of concentric circles is also set to 20, and the equal radian method is adopted for the sampling points in each circle of sampling track, which is 120 sampling points, as shown in Fig. 3. At the same time, the IEEE Instrumentation & Measurement Magazine 19

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