A five-axis machining continuous tool axis fairing method based on rotation vector interpolation

Abstract

The present application belongs to the technical field of five-axis NC machining of complex curved surface, and discloses a five-axis machining continuous tool axis smoothing method based on rotation vector interpolation. The tool posture at the key tool position is described as a unit quaternion; the unit quaternion is converted into a rotation vector by logarithmic operation to simplify the subsequent interpolation smoothing calculation, while retaining all the motion information of the original posture; in order to realize the C3 continuity of the tool axis motion, a quintic B-spline curve is used as an interpolation tool to accurately represent the intermediate rotation vector on the whole machining trajectory; in order to obtain the B-spline curve interpolated with the key tool posture, the energy functional minimization of the spline curve and the satisfaction of the key tool position constraint are comprehensively considered, a linear equation set based on the Lagrange multiplier method is constructed and solved to determine the control point coordinates of the B-spline curve; finally, the intermediate rotation vector obtained by interpolation is mapped back to the quaternion space by using the exponential operation of the quaternion, so as to determine the smooth intermediate tool posture.

Description

Technical Field

[0001] This invention belongs to the field of five-axis CNC machining technology for complex curved surfaces, and particularly relates to a continuous tool axis smoothing method for five-axis machining based on rotational vector interpolation. Background Technology

[0002] Currently, in many industrial sectors such as automotive and aerospace, the manufacturing of complex curved surface parts is particularly critical, and five-axis CNC machining technology is a crucial support for achieving high-performance manufacturing of such parts. Given the complexity of the geometry of these parts, the tilt angle of the cutting tool needs to be adaptively adjusted during five-axis machining to ensure no interference both locally and globally during the machining process. Simultaneously, due to the inherent limitations of machine tool drive capabilities, the smoothness of tool posture changes must be considered to eliminate severe tool axis oscillation and ensure a smooth and stable tool trajectory. Therefore, one of the challenges in five-axis machining trajectory planning lies in how to simultaneously achieve tool interference avoidance and a smooth tool axis motion trajectory. Scholars both domestically and internationally have proposed two approaches to this issue. One approach involves constructing a feasible space and then selecting the optimal tool axis vector within that space. For example, Ding Han et al.'s invention patent, "A Planning Method for Smooth, Interference-Free Tool Paths in Five-Axis CNC Machining" (patent number: CN200710045183.9), constructs discrete reachable direction cones at each tool position to ensure interference-free operation, and then selects the smooth tool axis from these cones. The other approach involves generating a smooth tool axis first and then adjusting it to avoid interference. The former requires calculating the feasible space at each tool position, which is computationally expensive. The latter often uses interpolation algorithms to generate intermediate tool postures, eliminating the need for tedious interference judgments at each tool position and each candidate tool axis. For example, the invention patent, "A Method for Generating Five-Axis Paths by Interpolating the Directions of Several Given Control Points" (patent number: CN201110439114.2), selects control points and corresponding tool axes on the machining path, calculates the elevation and azimuth angles corresponding to the tool axis at each control point, and uses linear interpolation to obtain the intermediate tool axis. Given the unique advantages of quaternions in representing the rotation and orientation of three-dimensional objects, such as faster speed, smooth interpolation, effective avoidance of gimbal lock-up, and smaller storage space, quaternion-based interpolation is often used in tool axis planning in five-axis machining. For example, the literature "Ho MC, Hwang YR, 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 tool axis smoothing method based on spherical linear interpolation, which aims to interpolate key tool postures to generate a smooth tool axis trajectory along the entire machining path. The invention patent "A method for smoothing the machining path of a five-axis machining tool" (patent number: CN201210157900.8) discloses a tool axis vector smoothing method based on quaternions.The invention patent "Spherical Machining Method for CNC Machine Tools Based on Quaternion Spiral Spherical Interpolation" (Patent No.: CN201711318701.X) proposes a spherical machining method using quaternion interpolation based on the spherical spiral method. However, most of the aforementioned quaternion-based tool axis smoothing methods are linear interpolations, which are difficult to guarantee high-order continuity of tool axis motion. Since rotation vectors are more compact in representing rotational motion than unit quaternions, the continuous tool axis smoothing method based on rotation vectors proposed in this invention has not been addressed by the aforementioned methods. To date, no five-axis machining continuous tool axis smoothing interpolation method based on rotation vectors and directly targeting tool axis smoothing has been reported. Summary of the Invention

[0003] To achieve continuous tool axis smoothing interpolation in five-axis machining and ensure the continuity of tool axis movement, this invention proposes a method for continuous tool axis smoothing in five-axis machining based on rotational vector interpolation.

[0004] The technical solution of this invention:

[0005] A five-axis machining continuous tool axis smoothing method based on rotational vector interpolation is proposed. First, quaternions are used as a mathematical tool to describe the tool posture at key tool positions as unit quaternions. Then, the logarithm of the unit quaternion is used to convert it into a rotational vector to simplify subsequent interpolation smoothing calculations, while preserving all motion information of the original posture. This method aims to achieve C-axis smoothing of the tool axis motion. 3 To ensure continuity, a quintic B-spline curve is used as an interpolation tool to accurately represent the intermediate rotation vector along the entire machining trajectory. To obtain the B-spline curve interpolated to the critical tool position posture, a system of linear equations based on the Lagrange multiplier method is constructed and solved, taking into account both the minimization of the energy functional of the spline curve and the satisfaction of the critical tool position constraints, to determine the coordinates of the control points of the B-spline curve. Finally, the intermediate rotation vector obtained by interpolation is mapped back to the quaternion space using the exponential operation of quaternions, thereby determining the smooth intermediate tool posture.

[0006] The specific steps are as follows:

[0007] Step 1) Considering the rotation of the tool axis vector relative to the initial orientation, convert the tool attitude at the critical tool position into a unit quaternion: Based on the pre-specified cutting characteristics, the interference-free critical tool position is set as follows: in For the knife point, o s Let be the three-dimensional tool axis vector in the workpiece coordinate system, s be the index of the critical tool position, and j be the index of the critical tool position. s Let be the index of the tool position point contained in the s-th critical tool position within the entire tool position point set; thus, the critical tool position... Tool axis vector o s The conversion process to a unit quaternion is represented as follows:

[0008]

[0009] In the formula, o r Indicates the initial orientation of the tool after tool setting, v s For o r Rotate to o s The unit axis of rotation, θ s For o r Rotate to o s The rotation angle, q s For o r Rotate to o s The corresponding unit quaternion; when o r =±o s The unit quaternion is specified using the following formula:

[0010]

[0011] Step 2) Using the logarithmic operation of unit quaternions, determine the unit quaternion corresponding to the critical tool axis. Convert to a rotation vector to simplify subsequent interpolation smoothing calculations:

[0012] R s =2ln (q) s )=θ s v s (3)

[0013] In the formula, R s unit quaternion q s The corresponding rotation vector is a three-dimensional vector;

[0014] Step 3) Perform smooth interpolation on the rotation vector obtained in Step 2); First, use a quintic B-spline curve as the interpolation tool to accurately represent the intermediate rotation vector on the entire machining trajectory:

[0015]

[0016] In the formula, Here are the control points for the quintic B-spline curve; k is the degree of the quintic B-spline curve, and k = 5; m+1 is the number of control points; N i,k (u) is a quintic B-spline basis function defined on the parameter u and the nodal vector U; to determine the nodal vector U, first, the tool point set is... Perform the following centripetal parameterization:

[0017]

[0018] In the formula, These are the two endpoints, the knife position p. c,0 , The parameterization results It is the intermediate knife point p c,j The parameterized result, where index j = 1,…,n c -1, n c +1 represents the total number of tool points; according to the parameterization result of equation (5) Node vector U = [u0, u1, ..., u m+k+1 Determined by the following formula:

[0019]

[0020] In the formula, u i Let i represent the i-th node of the node vector U, where i = 0, 1, ..., m+k+1. That is, the knife point p with index t. c,t The parameterization result is determined by equation (5), while a, t, and α are intermediate variables involved in the calculation of the node vector U, determined by equation (6). Subsequently, considering the minimization of the energy functional of the spline curve and the satisfaction of the key tool position constraints, the control points of the B-spline curve are obtained. The energy functional of the curve is defined as follows:

[0021]

[0022] To satisfy the critical tool position constraint, the B-spline curve described by equation (4) must pass through the critical tool position point. Corresponding rotation vector That is, it satisfies the following equation:

[0023]

[0024] In the formula, It is index j s The parameters of the tool position point are determined by equation (5); therefore, the interpolation task is transformed into the following optimization task:

[0025]

[0026] This is a quadratic minimization problem with linear constraints; by adding the Lagrange multiplier vector and the constraints shown in equation (8) to the objective function E, we obtain the following Lagrange function L, namely:

[0027]

[0028] In the formula, It is a Lagrange multiplier; let the partial derivatives and The expression is zero, where g = 0, 1, ..., m, and h = 0, 1, ..., n. k This yields the following system of linear equations:

[0029]

[0030] Where, Φ=[φ i,j ] (m+1)×(m+1) It is a symmetric matrix with the following elements:

[0031]

[0032] The specific representation of matrix Ω is as follows:

[0033]

[0034] r and Λ are column vectors composed of control points and Lagrange multipliers, respectively; η is a column vector containing the interpolated rotation vector; then, the control points are obtained by solving the linear equation (11), thereby obtaining the rotation vector smoothly interpolated at the key tool position;

[0035] Step 4) Using quaternion exponentiation, the interpolated intermediate rotation vector is mapped back to quaternion space, thereby determining the smooth intermediate tool orientation:

[0036]

[0037] In the formula, R is the intermediate rotation vector obtained by interpolation in step 3); q R It is the pure quaternion corresponding to the three-dimensional vector R, i.e., q R = [0, R]; q is the unit quaternion corresponding to the rotation of the intermediate tool orientation relative to the initial tool orientation; with the help of q, the intermediate tool orientation changes from the initial tool orientation o r Performing rotation q yields:

[0038]

[0039] In the formula, O r =[0,o r [This refers to the initial orientation of the cutting tool.] r The corresponding pure quaternion, q * It is the conjugate quaternion of q obtained by inverting the imaginary part of q. It is the intermediate tool posture The corresponding pure quaternion, extract the quaternion. The required intermediate tool orientation can be obtained from the vector part.

[0040] The beneficial effects of this invention are as follows: First, by utilizing the logarithmic operation of unit quaternions, the unit quaternion corresponding to the critical tool axis is converted into a more compact rotation vector form, simplifying the interpolation smoothing calculation; the intermediate rotation vector is described by a quintic B-spline curve, and by comprehensively considering the minimization of the energy functional of the spline curve and the satisfaction of the critical tool position constraints, the interpolated rotation vector satisfies C. 3 Continuity; Finally, the interpolated intermediate rotation vector is mapped back to the quaternion space. Since only quaternion exponentiation is involved between the two, the higher-order continuity of the rotation vector is preserved in the rotational motion represented by the quaternion, thus obtaining a smooth and higher-order continuous intermediate tool posture. The method of this invention provides tool axis smoothing based on rotation vector interpolation, derives the link process of mutual conversion between critical tool axes, quaternions, and rotation vectors, and provides a continuous interpolation method between critical tool axes using rotation vectors as intermediate auxiliary tools. The method is simple in form, and the continuity and smoothness of the interpolation results are higher than those of the traditional spherical linear interpolation method, realizing smooth tool movement in five-axis machining. Attached Figure Description

[0041] Figure 1 This is a flowchart of a five-axis machining continuous tool axis smoothing method based on rotational vector interpolation;

[0042] Figure 2 This is a schematic diagram of the initial orientation of an AC dual-rotor type five-axis CNC machine tool and its cutting tools;

[0043] Figure 3 This is a schematic diagram of the machining of curved surfaces and key tool positions;

[0044] Figure 4 It is the smooth tool axis vector obtained by interpolation using the proposed method;

[0045] Figure 5 It is the tool axis vector obtained by the traditional spherical linear interpolation method. Detailed Implementation

[0046] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0047] Figure 1 A flowchart of a five-axis machining continuous tool axis smoothing method based on rotation vector interpolation is presented. Based on this method, the tool position set and key tool positions are input; the tool position set is parameterized, and node vectors are calculated; the key tool positions are converted into unit quaternions, and the unit quaternions are converted into rotation vectors; a system of linear equations based on the Lagrange multiplier method is derived; the intermediate rotation vectors are inversely mapped back to the quaternion space; finally, a high-order continuous intermediate tool axis vector is output.

[0048] For example Figure 2 Taking the AC dual rotary table type five-axis CNC machine tool shown as an example

[0049] (1) As follows Figure 3 The tool orientation at the critical tool position shown is converted to a unit quaternion: Let the interference-free critical tool position, set according to the pre-specified cutting characteristics, be... in For the knife point, o s Let be the three-dimensional tool axis vector in the workpiece coordinate system, s be the index of the critical tool position, and j be the index of the critical tool position. s Let be the index of the s-th critical tool position point in the entire tool position point set; thus, the critical tool position... Tool axis vector o s The conversion process to a unit quaternion is represented as follows:

[0050]

[0051] In the formula, o r This indicates the initial orientation of the tool after tool setting. For an AC dual-rotor type five-axis CNC machine tool, o r =[0,0,1], v s For o r Rotate to o s The unit axis of rotation, θ s For o r Rotate to o s The rotation angle, q s For o r Rotate to o s The corresponding unit quaternion. For the special case o... r =±o s Then, the following formula is used to specify it:

[0052]

[0053] (2) The unit quaternion corresponding to the critical tool axis Convert to a rotation vector:

[0054] R s =2ln (q) s )=θ s v s (3)

[0055] In the formula, R s unit quaternion q s The corresponding rotation vector is a three-dimensional vector.

[0056] (3) Perform smooth interpolation on the obtained rotation vectors. First, use a quintic B-spline curve to accurately represent the intermediate rotation vectors on the entire machining trajectory:

[0057]

[0058] In the formula, Let m be the control points of the quintic B-spline curve; k is the degree of the B-spline curve, and k = 5; m+1 is the number of control points; N i,k (u) is a B-spline basis function defined on the parameter u and the node vector U; to determine the node vector U, first, the tool point set is... Perform the following centripetal parameterization:

[0059]

[0060] Based on the parameterization results of equation (5) Node vector U = [u0, u1, ..., u m+k+1 It can be determined by the following formula:

[0061]

[0062] Subsequently, considering both the minimization of the energy functional of the spline curve and the satisfaction of the critical tool position constraints, the control points of the B-spline curve are determined. The energy functional of this curve is defined as follows:

[0063]

[0064] Satisfying the critical tool position constraint means that the B-spline curve described by equation (4) must pass through the critical tool position point. Corresponding rotation vector That is, it satisfies the following equation:

[0065]

[0066] Therefore, the smooth interpolation task is transformed into the following optimization problem:

[0067]

[0068] By adding Lagrange multiplier vectors and constraints to the objective function E, the following Lagrange function is obtained:

[0069]

[0070] in It is a Lagrange multiplier. Let the partial derivatives... and The expression is zero, where g = 0, 1, ..., m, and h = 0, 1, ..., n. k We can obtain the following system of linear equations:

[0071]

[0072] Where Φ=[φ i,j ](m+1)×(m+1) It is a symmetric matrix with the following elements:

[0073]

[0074] The specific representation of matrix Ω is as follows:

[0075]

[0076] r and Λ are column vectors composed of control points and Lagrange multipliers, respectively. η is a column vector containing the interpolated rotation vector. Subsequently, the control points are obtained by solving the linear equation (11), from which the rotation vector smoothly interpolated at the critical tool position can be obtained.

[0077] (4) Map the interpolated intermediate rotation vector back to quaternion space to determine the smooth intermediate tool posture:

[0078]

[0079] In the formula, r is the intermediate rotation vector obtained through interpolation; q R It is the pure quaternion corresponding to the three-dimensional vector r, i.e., q R = [0, r]; q is the unit quaternion corresponding to the rotation of the intermediate tool orientation relative to the initial tool orientation. Therefore, with the help of q, the intermediate tool orientation can be determined from the initial tool orientation o. r By performing a rotation q, we obtain:

[0080]

[0081] In the formula, O r =[0,o r [This refers to the initial orientation of the cutting tool.] r The corresponding pure quaternion, for an AC dual-rotor type five-axis CNC machine tool, is O r =[0,0,0,1], q * It is the conjugate quaternion of q obtained by inverting the imaginary part of q. It is the intermediate tool posture The corresponding pure quaternion, extract the quaternion.

[0082] The required intermediate tool orientation can be obtained from the vector part, such as Figure 4 As shown. (Attached) Figure 5 This is the tool axis vector obtained by the traditional spherical linear interpolation method under the same critical tool position settings. Figure 4 The tool axis vector determined by the method of the present invention is shown in the figure. Comparing the two, it can be seen that the tool axis motion determined by the method of the present invention has higher continuity and better smoothness at the key tool position.

Claims

1. A five-axis machining continuous tool axis smoothing method based on rotational vector interpolation, characterized in that, First, quaternions are used as a mathematical tool to describe the tool attitude at critical tool positions as unit quaternions. Then, logarithmic operations on the unit quaternions are used to convert them into rotation vectors to simplify subsequent interpolation and smoothing calculations, while preserving all motion information of the original attitude. This is to achieve the C-axis motion of the tool axis. 3 To ensure continuity, a quintic B-spline curve is used as an interpolation tool to accurately represent the intermediate rotation vector along the entire machining trajectory. To obtain the B-spline curve interpolated to the critical tool position posture, a linear equation system based on the Lagrange multiplier method is constructed and solved, taking into account both the minimization of the energy functional of the spline curve and the satisfaction of the critical tool position constraints, to determine the coordinates of the control points of the B-spline curve. Finally, the intermediate rotation vector obtained by interpolation is mapped back to the quaternion space using the exponential operation of quaternions, thereby determining the smooth intermediate tool posture. The specific steps are as follows: Step 1) Considering the rotation of the tool axis vector relative to the initial orientation, convert the tool attitude at the critical tool position into a unit quaternion: Based on the pre-specified cutting characteristics, the interference-free critical tool position is set as follows: ,in For the knife point, The three-dimensional tool axis vector in the workpiece coordinate system. This is the sequence number of the critical tool position. For the first Each critical tool position contains the index of the tool position point within the entire set of tool position points; thus, the critical tool position... Tool axis vector at the location The conversion process to a unit quaternion is represented as follows: （1） In the formula, This indicates the initial position of the tool after tool setting is completed. For the reason Rotate to The unit axis of rotation, For the reason Rotate to rotation angle, For the reason Rotate to The corresponding unit quaternion; when The unit quaternion is specified using the following formula: （2） Step 2) Using the logarithmic operation of unit quaternions, determine the unit quaternion corresponding to the critical tool axis. Convert to a rotation vector to simplify subsequent interpolation smoothing calculations: （3） In the formula, unit quaternion The corresponding rotation vector is a three-dimensional vector; Step 3) Perform smooth interpolation on the rotation vector obtained in Step 2); First, use a quintic B-spline curve as the interpolation tool to accurately represent the intermediate rotation vector on the entire machining trajectory: （4） In the formula, These are the control points for the quintic B-spline curve; Let be the degree of the quintic B-spline curve, and ; It is the number of control points; It is a quintic B-spline basis function, defined in the parameter and node vector Above; to determine the node vector First, the tool point set Perform the following centripetal parameterization: （5） In the formula, , These are the two endpoints of the blade. , The parameterization results It is the intermediate knife point. The parameterized result, where the index , This is the total number of tool points; according to the parameterization result of equation (5) Node vectors Determined by the following formula: （6） In the formula, Representative node vector The 1 node , The index is knife point The parameterization result is determined by equation (5), and , , It is to calculate the node vector The intermediate variables involved are determined by equation (6); subsequently, considering both the minimization of the energy functional of the spline curve and the satisfaction of the critical tool position constraints, the control points of the B-spline curve are obtained; the energy functional of the curve is defined as follows: （7） To satisfy the critical tool position constraint, the B-spline curve described by equation (4) must pass through the critical tool position point. Corresponding rotation vector That is, it satisfies the following equation: （8） In the formula, Is the index as The parameters of the tool position point are determined by equation (5); therefore, the interpolation task is transformed into the following optimization task: （9） This is a quadratic minimization problem with linear constraints; by optimizing the objective function... Adding the Lagrange multiplier vectors and the constraints shown in equation (8), we obtain the following Lagrange function. ,Right now: （10） In the formula It is a Lagrange multiplier; let the partial derivatives and It is zero, of which , This yields the following system of linear equations: （11） in, It is a symmetric matrix with the following elements: （12） matrix The specific representation is as follows: （13） and These are column vectors composed of control points and Lagrange multipliers, respectively. It is a column vector containing the interpolated rotation vector; then, the control points are obtained by solving the linear equation (11), thereby obtaining the rotation vector smoothly interpolated at the key tool position; Step 4) Using quaternion exponentiation, the interpolated intermediate rotation vector is mapped back to quaternion space, thereby determining the smooth intermediate tool orientation: （14） In the formula, It is the intermediate rotation vector obtained by interpolation in step 3); It is a three-dimensional vector The corresponding pure quaternion, i.e. ; It is the unit quaternion corresponding to the rotation of the intermediate tool orientation relative to the initial tool position; using The intermediate tool orientation changes from the initial tool position. Implement rotation get: （15） In the formula, The initial orientation of the cutting tool The corresponding pure quaternion, Through the Inverting the imaginary part yields The conjugate quaternion, It is the intermediate tool posture The corresponding pure quaternion, extract the quaternion. The required intermediate tool orientation can be obtained from the vector part.

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