CN-119148616-B - Five-axis processing continuous cutter shaft fairing method based on rotation vector interpolation

Abstract

The invention belongs to the technical field of five-axis numerical control machining of complex curved surfaces, and discloses a five-axis continuous cutter shaft fairing method based on rotation vector interpolation. The method comprises the steps of describing the tool posture of a key tool position as a unit quaternion, converting the unit quaternion into a rotation vector by utilizing logarithmic operation of the unit quaternion to simplify subsequent interpolation fairing calculation and simultaneously reserve all motion information of an original posture, adopting a quintic B spline curve as an interpolation tool to accurately represent an intermediate rotation vector on the whole processing track for realizing C3 continuity of the tool shaft movement, comprehensively considering energy functional minimization of the spline curve and satisfaction of key tool position constraint for obtaining the B spline curve interpolated on the key tool position posture, constructing and solving a linear equation set based on a Lagrange multiplier method to determine control point coordinates of the B spline curve, and finally mapping the intermediate rotation vector obtained by interpolation back to a quaternion space by utilizing exponential operation of the quaternion so as to determine the smooth intermediate tool posture.

Inventors

• XU JINTING
• WU LEI
• SUN YUWEN

Assignees

• 大连理工大学

Dates

Publication Date

20260508

Application Date

20240914

Claims (1)

1. 1. A five-axis machining continuous cutter shaft fairing method based on rotary vector interpolation is characterized by comprising the steps of firstly, describing a cutter gesture at a key cutter position as a unit quaternion by adopting a quaternion mathematical tool, then, converting the cutter gesture into a rotary vector by utilizing logarithmic operation of the unit quaternion to simplify subsequent interpolation fairing calculation and simultaneously reserve all motion information of an original gesture, accurately representing an intermediate rotary vector on the whole machining track by adopting a quintic B-spline curve as an interpolation tool for realizing C 3 continuity of cutter shaft motion, constructing and solving a linear equation set based on a Lagrangian multiplier method by comprehensively considering energy functional minimization and key cutter position constraint satisfaction of the spline curve to determine a control point coordinate of the B-spline curve, and finally, mapping the intermediate rotary vector obtained by interpolation into a quaternion space by utilizing exponential operation of the quaternion so as to determine a smooth intermediate cutter gesture; the method comprises the following specific steps: Step 1), converting the tool posture of the key tool position into a unit quaternion by considering the rotation of the cutter shaft vector relative to the initial azimuth, wherein the key tool position without interference set according to the preset cutting characteristics is as follows Wherein As the position point of the knife, the position of the knife, Is a three-dimensional cutter shaft vector in a workpiece coordinate system, Is the serial number of the key tool bit, Is the first The key tool positions comprise the serial numbers of the tool positions concentrated in the whole tool positions, so that the key tool positions Arbor vector at The conversion process into unit quaternions is expressed as: (1) in the formula, Indicating the initial orientation of the cutter after the cutter is aligned, Is composed of Rotated to Is provided with a unit rotation axis of (a), Is composed of Rotated to Is provided with a rotation angle of (c), Is composed of Rotated to Corresponding unit quaternion when The unit quaternion is specified using the following formula: (2) Step 2), utilizing logarithmic operation of unit quaternions to make the unit quaternions corresponding to the key cutter shaft Conversion to a rotation vector to simplify subsequent interpolation fairing calculations: (3) in the formula, Is a unit quaternion The corresponding rotation vector is a three-dimensional vector; Step 3), carrying out fairing interpolation on the rotation vector obtained in the step 2), and firstly, adopting a quintic B spline curve as an interpolation tool to accurately represent the intermediate rotation vector on the whole processing track: (4) in the formula, Control points for the cubic B spline curve; Is the number of times of the quintic B-spline curve, and ; Is the number of control points; is a cubic B-spline basis function defined in parameters Sum node vector To determine the node vector First, tool setting position point set Centripetal parameterization was performed as follows: (5) in the formula, 、 Respectively two end knife sites 、 Is used for the parameter results of the (a), Is the middle knife site Where the index is , Then the total number of knife positions, parameterized result according to equation (5) Node vector Determined by the following formula: (6) in the formula, Representative node vector Is the first of (2) The number of nodes in the network is, , Then is indexed as Tool position point of (2) Is determined by the formula (5) 、 、 Is the calculation of the node vector The related intermediate variables are determined by a formula (6), then, the energy functional minimization of the spline curve and the satisfaction of the key tool position constraint are comprehensively considered, the control point of the B spline curve is obtained, and the energy functional of the curve is defined as follows: (7) if it is desired to satisfy the critical tool position constraint, the B-spline curve described by equation (4) must pass through the critical tool position points Corresponding rotation vector I.e. the following equation is satisfied: (8) in the formula, Is indexed as The parameters of the knife sites of (2) are determined by equation (5), and thus, the interpolation task is converted into the following optimization task: (9) This is a quadratic minimization problem with linear constraints by optimizing the objective function Adding Lagrangian multiplier vector and constraint condition shown in (8) to obtain Lagrangian function as follows The method comprises the following steps: (10) In the middle of Is Lagrangian multiplier, let partial derivative And Zero, where , The following linear system of equations is obtained: (11) Wherein, the Is a symmetric matrix, the elements of which are: (12) Matrix array The specific expression of (2) is as follows: (13) And Respectively a column vector consisting of a control point and a Lagrangian multiplier; Then solving the linear equation (11) to obtain a control point, thereby obtaining a rotation vector smoothly interpolated at the critical tool position; Step 4), mapping the intermediate rotation vector obtained by interpolation back to a quaternion space by utilizing the exponential operation of the quaternion, so as to determine a smooth intermediate tool posture: (14) in the formula, The intermediate rotation vector obtained by interpolation in the step 3); is a three-dimensional vector Corresponding pure quaternions, i.e. ; Is a unit quaternion corresponding to the rotation of the posture of the middle cutter relative to the initial direction of the cutter by means of The middle tool posture is determined by the initial orientation of the tool By effecting rotation The method comprises the following steps: (15) in the formula, Is the initial orientation of the tool The corresponding pure quaternion is used for the conversion of the data, Is by means of alignment of Is obtained by inverting the imaginary part of (2) Is used for the matching of the conjugated quaternion of (c), Is the middle tool posture Corresponding pure quaternion, extracting quaternion The desired intermediate tool pose is obtained.

Description

Five-axis processing continuous cutter shaft fairing method based on rotation vector interpolation Technical Field The invention belongs to the technical field of five-axis numerical control machining of complex curved surfaces, and particularly relates to a five-axis continuous cutter shaft fairing method based on rotation vector interpolation. Background At present, in various industrial fields such as automobiles, aviation and the like, the manufacture of complex curved surface parts is particularly critical, and a five-axis numerical control processing technology is an important support for realizing the high-performance manufacture of the parts. In view of the complexity of such part geometry, the inclination angle of the cutting tool needs to be adaptively adjusted during five-axis machining to ensure that the machining process has no interference between local and global. Meanwhile, due to inherent limitation of the driving capability of the machine tool, the smoothness of the posture change of the cutter must be considered so as to eliminate the severe swing of the cutter shaft and ensure the smooth and stable movement track of the cutter. Therefore, one of the difficulties in five-axis machining track planning is how to simultaneously realize the avoidance of tool interference and the fairing of the cutter shaft motion track. Two ideas exist for the problem of the students at home and abroad, one is to construct a feasible space and optimize cutter shaft vectors in the feasible space, for example, the 'five-axis numerical control processing smooth and interference-free cutter path planning method' of patent No. Ding Han et al (patent No. CN 200710045183.9) is to construct discrete reachable direction cones at each cutter site to ensure interference-free, and then optimizing smooth cutter shafts from the reachable direction cones, and the other is to firstly generate smooth cutter shafts and then adjust the smooth cutter shafts to avoid interference. The former needs to calculate the feasible space at each tool position, the calculation cost is high, and the latter often generates the middle tool posture by means of an interpolation algorithm, so that complicated interference judgment of tool positions and alternative tool shafts is not needed. For example, the invention patent 'a five-axis path generating method for interpolating a plurality of given control point directions' (patent number: CN 201110439114.2) selects a control point and a corresponding cutter shaft on a processing path, calculates an elevation angle and an azimuth angle corresponding to the cutter shaft at each control point, and obtains an intermediate cutter shaft by using a linear interpolation method. In view of the unique advantages of quaternions in representing the rotation and orientation of three-dimensional objects, such as faster speeds, providing smooth interpolation, effectively avoiding gimbal lock problems, smaller storage space, etc., quaternion-based interpolation is also often used in arbor planning in five-axis machining. For example, document "Ho M C,Hwang Y R,Hu C H.Five-axis tool orientation smoothing using quaternion interpolation algorithm[J].International Journal of Machine Tools and Manufacture,2003,43(12):1259-1267." discloses a knife axis fairing method based on a spherical linear interpolation method, aiming at interpolating key tool poses to generate a straightened knife axis track on the whole processing path. The invention patent (a method for smoothing a machining path of a five-axis machining tool) (patent number: CN 201210157900.8) discloses a cutter shaft vector fairing method based on quaternion. The invention patent 'a numerical control machine tool spherical surface processing method based on a quaternion spiral line spherical surface interpolation method' (patent number: CN201711318701. X) is based on a spherical surface spiral line method, and proposes a spherical surface processing method adopting quaternion interpolation. However, the cutter shaft fairing method based on quaternion is mostly linear interpolation, and high-order continuity of cutter shaft movement is difficult to ensure. The form of the rotation vector relative to the unit quaternion is more compact when the rotation vector represents the rotation motion, the continuous cutter shaft fairing method based on the rotation vector is not involved, and a five-axis machining continuous cutter shaft fairing interpolation method based on the rotation vector and directly aiming at the cutter shaft fairing is not reported so far. Disclosure of Invention In order to realize five-axis processing continuous cutter shaft fairing interpolation and ensure the continuity of cutter shaft movement, the invention provides a five-axis processing continuous cutter shaft fairing method based on rotation vector interpolation. The technical scheme of the invention is as follows: A five-axis machining continuous cutter shaft fairing method bas