Five-axis machining continuous cutter axis fairing method based on rotating vector interpolation

Abstract

The invention belongs to the technical field of complex curved surface five-axis numerical control machining, and discloses a five-axis machining continuous cutter axis fairing method based on rotating vector interpolation. Describing the tool posture at the key tool location as a unit quaternion; the unit quaternion is converted into a rotating vector through logarithm operation of the unit quaternion so as to simplify follow-up interpolation fairing calculation, and meanwhile all motion information of an original posture is reserved; in order to realize C3 continuity of cutter shaft motion, a quintic B spline curve is used as an interpolation tool to accurately represent a middle rotation vector on a whole processing track; in order to obtain a B-spline curve interpolated in the key cutter position attitude, energy functional minimization of the spline curve and satisfaction of key cutter position constraints are comprehensively considered, and a linear equation set based on a Lagrange multiplier method is constructed and solved so as to determine control point coordinates of the B-spline curve; finally, the intermediate rotation vector obtained through interpolation is mapped back to a quaternion space through quaternion exponential operation, and therefore the smooth intermediate cutter posture is determined.

Description

Technical Field

[0001] The invention belongs to the technical field of five-axis CNC machining of complex surfaces, and in particular relates to a five-axis machining continuous tool axis smoothing method based on rotating vector interpolation. Background Art

[0002] At present, the manufacturing of complex curved parts is particularly critical in many industrial fields such as automobiles and aviation, and five-axis CNC machining technology is an important support for achieving high-performance manufacturing of such parts. In view of the complexity of the geometric shapes of such parts, the inclination angle of the cutting tool needs to be adaptively adjusted during the five-axis machining process to ensure that there is no interference both locally and globally during the machining process. At the same time, due to the inherent limitations of the machine tool's driving capabilities, the smoothness of the tool posture changes must be considered to eliminate the violent swing of the tool axis and ensure that the tool's motion trajectory is smooth and stable. Therefore, one of the difficulties in five-axis machining trajectory planning is how to simultaneously achieve the avoidance of tool interference and the smoothness of the tool axis motion trajectory. Scholars at home and abroad have two ideas about this. One is to construct a feasible space and optimize the tool axis vector in the feasible space. For example, Ding Han et al.'s invention patent "A method for planning a smooth and interference-free tool path for five-axis CNC machining" (patent number: CN200710045183.9) constructs a discrete reachable direction cone at each tool position to ensure no interference, and then optimizes the smooth tool axis from the reachable direction cone; the other idea is to first generate a smooth tool axis and then adjust it to avoid interference. The former requires calculating the feasible space at each tool position, which is computationally expensive, while the latter often uses an interpolation algorithm to generate an intermediate tool posture without the tedious interference judgment of each tool position and each alternative tool axis. For example, the invention patent "A method for generating a five-axis path 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 corresponding to the tool axis at each control point, and uses a linear interpolation method 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 universal lock problems, and smaller storage space, quaternion-based interpolation is often used in tool axis planning in five-axis machining. For example, the document "Ho MC, Hwang YR, Hu CH. 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 method, which aims to interpolate key tool postures to generate a smooth tool axis trajectory on 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 "CNC machine tool spherical processing method based on quaternion helix spherical interpolation method" (patent number: CN201711318701.X) is based on the spherical helix method, and proposes a spherical processing method using quaternion interpolation. However, the above-mentioned tool axis smoothing methods based on quaternions are mostly linear interpolation, which makes it difficult to ensure the high-order continuity of the tool axis motion. Since the rotation vector is more compact than the unit quaternion when representing rotational motion, the above-mentioned methods for continuous tool axis smoothing based on the rotation vector proposed in the present invention are not involved. So far, a five-axis machining continuous tool axis smoothing interpolation method based on the rotation vector and directly targeting tool axis smoothing has not been reported. Summary of the invention

[0003] In order to realize continuous tool axis smoothing interpolation in five-axis machining and ensure the continuity of tool axis motion, the present invention proposes a continuous tool axis smoothing method for five-axis machining based on rotation vector interpolation.

[0004] The technical solution of the present invention:

[0005] A continuous tool axis smoothing method for five-axis machining based on rotation vector interpolation is proposed. First, the tool posture at the key tool position is described as a unit quaternion using the mathematical tool quaternion. Then, the unit quaternion is converted into a rotation vector using the logarithmic operation to simplify the subsequent interpolation smoothing calculation while retaining all the motion information of the original posture. 3 Continuity, quintic B-spline curve is used as interpolation tool to accurately represent the intermediate rotation vector on the entire machining trajectory; in order to obtain the B-spline curve interpolated on the key tool position posture, the energy functional minimization of the spline curve and the satisfaction of the key tool position constraints are comprehensively considered, and a linear equation group based on the Lagrange multiplier method is constructed and solved to determine the coordinates of the control points of the B-spline curve; finally, the exponential operation of quaternion is used to map the intermediate rotation vector obtained by interpolation back to the quaternion space, so as to determine 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, the tool posture at the key tool position is converted into a unit quaternion: The non-interference key tool position set according to the pre-specified cutting characteristics is in is the knife point, o s is the three-dimensional tool axis vector in the workpiece coordinate system, s is the serial number of the key tool position, j is s is the sequence number of the tool position point contained in the sth key tool position in the entire tool position point set; thus, the key tool position The knife axis vector o at s The conversion process to unit quaternion is expressed as:

[0008]

[0009] In the formula, o r Indicates the initial position of the tool after tool setting, v s For r Rotate to o s The unit rotation axis, θ s For r Rotate to o s The rotation angle, q s For 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) Use the logarithm operation of the unit quaternion to convert the unit quaternion corresponding to the key 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 is the 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 the quintic B-spline curve as an interpolation tool to accurately represent the intermediate rotation vector on the entire processing trajectory:

[0015]

[0016] In the formula, is the control point of 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 the quintic B-spline basis function, defined on the parameter u and the node vector U. To determine the node vector U, firstly align the tool position set Perform the following centripetal parameterization:

[0017]

[0018] In the formula, They are the two end point knife positions p c,0 , The parameterization result is is the middle knife point p c,j The parameterization result of c -1,n c +1 is the total number of tool positions; according to the parameterization result of formula (5) Node vector U=[u0,u1,…,u m+k+1 ] is determined by the following formula:

[0019]

[0020] In the formula, u i represents the i-th node of the node vector U, i=0,1,…,m+k+1, is the knife point p with index t c,t The parameterized result is determined by formula (5), while a, t, α are the intermediate variables involved in calculating the node vector U, which are determined by formula (6); then, the control points of the B-spline curve are obtained by comprehensively considering the minimization of the energy functional of the spline curve and the satisfaction of the key tool position constraints; the energy functional of the curve is defined as follows:

[0021]

[0022] If the critical tool position constraint is to be satisfied, the B-spline curve described by equation (4) must pass through the critical tool position point The corresponding rotation vector That satisfies the following equation:

[0023]

[0024] In the formula, is indexed by j s The parameters of the tool position are determined by formula (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 optimization objective function E, the following Lagrangian function L is obtained, namely:

[0027]

[0028] In the formula, is the Lagrange multiplier; let the partial derivative and is zero, where g = 0, 1, ..., m, h = 0, 1, ..., n k , we get the following linear equations:

[0029]

[0030] Where Φ=[φ i,j ] (m+1)×(m+1) is a symmetric matrix whose elements are:

[0031]

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

[0033]

[0034] r and Λ are column vectors consisting 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 critical tool position;

[0035] Step 4) Use the quaternion exponential operation to map the interpolated intermediate rotation vector back to the quaternion space to determine the smooth intermediate tool posture:

[0036]

[0037] Where R is the intermediate rotation vector obtained by interpolation in step 3); R is the pure quaternion corresponding to the three-dimensional vector R, that is, q R = [0, R]; q is the unit quaternion corresponding to the rotation of the intermediate tool posture relative to the initial tool orientation; with the help of q, the intermediate tool posture is transformed from the initial tool orientation o r Applying the rotation q yields:

[0038]

[0039] In the formula, O r =[0,o r ] is the initial orientation of the tool o r The corresponding pure quaternion, q * is the conjugate quaternion of q obtained by negating the imaginary part of q, The middle tool posture Corresponding pure quaternion, extract quaternion The vector part of can get the required intermediate tool posture.

[0040] The beneficial effects of the present invention are as follows: firstly, by means of the logarithmic operation of the unit quaternion, the unit quaternion corresponding to the key tool axis is converted into a more compact rotation vector form, thereby simplifying the interpolation smoothing calculation; the intermediate rotation vector is described by a quintic B-spline curve, and the energy functional minimization of the spline curve and the satisfaction of the key tool position constraint are comprehensively considered, so that the rotation vector obtained by interpolation satisfies C 3 Continuity; finally, the intermediate rotation vector obtained by interpolation is mapped back to the quaternion space. Since the two only involve the exponential operation of the quaternion, the high-order continuity of the rotation vector can be correspondingly retained to the rotational motion represented by the quaternion, thereby obtaining a smooth and high-order continuous intermediate tool posture. The method of the present invention provides tool axis smoothing based on rotation vector interpolation, deduces the link process from the key tool axis to the quaternion and the mutual conversion between the rotation vector, and provides a continuous interpolation method between the key tool axes with the help of the rotation vector as an intermediate auxiliary tool. The form is concise, and the continuity and smoothness of the interpolation result are higher than the traditional spherical linear interpolation method, realizing the smooth motion of the tool during five-axis machining. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 It is a flow chart of a continuous tool axis smoothing method for five-axis machining based on rotation vector interpolation;

[0042] Figure 2 It is a schematic diagram of the AC double rotary table five-axis CNC machine tool and its tool initial orientation;

[0043] Figure 3 It is a schematic diagram of the machining surface and key tool positions;

[0044] Figure 4 is the smoothing 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 DESCRIPTION

[0046] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

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

[0048] As Figure 2 The AC double rotary table five-axis CNC machine tool shown in the figure is an example

[0049] (1) Figure 3 The tool posture at the key tool position shown is converted into a unit quaternion: Assume that the non-interference key tool position set according to the pre-specified cutting characteristics is in is the knife point, o s is the three-dimensional tool axis vector in the workpiece coordinate system, s is the serial number of the key tool position, j is s is the sequence number of the sth key tool position in the entire tool position set; thus, the key tool position The knife axis vector o at s The conversion process to unit quaternion is expressed as:

[0050]

[0051] In the formula, o r Indicates the initial position of the tool after tool setting. For AC dual-rotary five-axis CNC machine tools, o r =[0,0,1], v s For r Rotate to o s The unit rotation axis, θ s For r Rotate to o s The rotation angle, q s For 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 key tool axis Convert to a rotation vector:

[0054] R s =2ln (q s )=θ s v s (3)

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

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

[0057]

[0058] In the formula, is the control point 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 the B-spline basis function, defined on the parameter u and the node vector U. To determine the node vector U, firstly align the tool position set Perform the following centripetal parameterization:

[0059]

[0060] According to the parameterization results of formula (5) Node vector U=[u0,u1,…,u m+k+1 ] can be determined by the following formula:

[0061]

[0062] Then, the control points of the B-spline curve are obtained by comprehensively considering the minimization of the energy functional of the spline curve and the satisfaction of the key tool position constraints. The energy functional of the 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 The corresponding rotation vector That satisfies the following equation:

[0065]

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

[0067]

[0068] By adding the Lagrangian multiplier vector and constraints to the optimization objective function E, the following Lagrangian function is obtained:

[0069]

[0070] in is the Lagrange multiplier. Let the partial derivative be and is zero, where g = 0, 1, ..., m, h = 0, 1, ..., n k , we can get the following linear equations:

[0071]

[0072] where Φ=[φ i,j ](m+1)×(m+1) is a symmetric matrix whose elements are:

[0073]

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

[0075]

[0076] r and Λ are column vectors consisting 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), 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 the quaternion space to determine the smooth intermediate tool posture:

[0078]

[0079] Where r is the intermediate rotation vector obtained by interpolation; q R is the pure quaternion corresponding to the three-dimensional vector r, that is, q R = [0, r]; q is the unit quaternion corresponding to the rotation of the intermediate tool posture relative to the initial tool orientation. Therefore, with the help of q, the intermediate tool posture can be obtained from the initial tool orientation o r Implementing the rotation q yields:

[0080]

[0081] In the formula, O r =[0,o r ] is the initial orientation of the tool o r The corresponding pure quaternion, for AC dual-rotary table five-axis CNC machine tools, O r =[0,0,0,1],q * is the conjugate quaternion of q obtained by negating the imaginary part of q, The middle tool posture Corresponding pure quaternion, extract quaternion

[0082] The vector part of can get the required intermediate tool posture, such as Figure 4 Attached Figure 5 is the tool axis vector obtained by the traditional spherical linear interpolation method under the same key tool position setting, Figure 4 This is the tool axis vector determined by the method of the present invention. By 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 rotation vector interpolation, characterized in that: Firstly, the tool posture at the key tool position is described as a unit quaternion using the mathematical tool of quaternion. Then, the unit quaternion is converted into a rotation vector by using the logarithmic operation to simplify the subsequent interpolation and smoothing calculation while retaining all the motion information of the original posture. 3 Continuity, using quintic B-spline curve as interpolation tool to accurately represent the intermediate rotation vector on the entire machining trajectory; in order to obtain the B-spline curve interpolated on the key tool position posture, the energy functional minimization of the spline curve and the satisfaction of the key tool position constraints are comprehensively considered, and a linear equation system based on the Lagrange multiplier method is constructed and solved to determine the coordinates of the control points of the B-spline curve; finally, using the exponential operation of quaternion, the intermediate rotation vector obtained by interpolation is mapped back to the quaternion space, so as to determine 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, the tool posture at the key tool position is converted into a unit quaternion: The non-interference key tool position set according to the pre-specified cutting characteristics is in is the knife point, o s is the three-dimensional tool axis vector in the workpiece coordinate system, s is the serial number of the key tool position, j is s is the sequence number of the tool position point contained in the sth key tool position in the entire tool position point set; thus, the key tool position The knife axis vector o at s The conversion process to unit quaternion is expressed as: In the formula, o r Indicates the initial position of the tool after tool setting, v s For r Rotate to o s The unit rotation axis, θ s For r Rotate to o s The rotation angle, q s For r Rotate to o s The corresponding unit quaternion; when o r = ±o s , the unit quaternion is specified using the following formula: Step 2) Use the logarithm operation of the unit quaternion to convert the unit quaternion corresponding to the key tool axis Convert to a rotation vector to simplify subsequent interpolation smoothing calculations: R s =2ln(q s )=θ s v s (3) In the formula, R s is the unit quaternion q s The corresponding rotation vector is a three-dimensional vector; Step 3) Perform smooth interpolation on the rotation vector obtained in step 2); first, use the quintic B-spline curve as an interpolation tool to accurately represent the intermediate rotation vector on the entire processing trajectory: In the formula, is the control point of 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 the quintic B-spline basis function, defined on the parameter u and the node vector U. To determine the node vector U, firstly align the tool position set Perform the following centripetal parameterization: In the formula, They are the two end point knife positions p c,0 , The parameterization result is is the middle knife point p c,j The parameterization result of c -1,n c +1 is the total number of tool positions; according to the parameterization result of formula (5) Node vector U=[u0,u1,…,u m+k+1 ] is determined by the following formula: In the formula, u i represents the i-th node of the node vector U, i=0,1,…,m+k+1, is the knife point p with index t c,t The parameterized result is determined by formula (5), while a, t, α are the intermediate variables involved in calculating the node vector U, which are determined by formula (6); then, the control points of the B-spline curve are obtained by comprehensively considering the minimization of the energy functional of the spline curve and the satisfaction of the key tool position constraints; the energy functional of the curve is defined as follows: If the critical tool position constraint is to be satisfied, the B-spline curve described by equation (4) must pass through the critical tool position point The corresponding rotation vector That satisfies the following equation: In the formula, is indexed by j s The parameters of the tool position are determined by formula (5); therefore, the interpolation task is transformed into the following optimization task: This is a quadratic minimization problem with linear constraints. By adding the Lagrange multiplier vector and the constraints shown in equation (8) to the optimization objective function E, the following Lagrangian function L is obtained, namely: In the formula, is the Lagrange multiplier; let the partial derivative and is zero, where g = 0, 1, ..., m, h = 0, 1, ..., n k , we get the following linear equations: Where Φ=[φ i,j ] (m+1)×(m+1) is a symmetric matrix whose elements are: The specific representation of the matrix Ω is as follows: r and Λ are column vectors consisting 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 critical tool position; Step 4) Use the quaternion exponential operation to map the interpolated intermediate rotation vector back to the quaternion space to determine the smooth intermediate tool posture: Where R is the intermediate rotation vector obtained by interpolation in step 3); R is the pure quaternion corresponding to the three-dimensional vector R, that is, q R = [0, R]; q is the unit quaternion corresponding to the rotation of the intermediate tool posture relative to the initial tool orientation; with the help of q, the intermediate tool posture is transformed from the initial tool orientation o r Applying the rotation q yields: In the formula, O r =[0,o r ] is the initial orientation of the tool o r The corresponding pure quaternion, q * is the conjugate quaternion of q obtained by negating the imaginary part of q, The middle tool posture Corresponding pure quaternion, extract quaternion The vector part of can get the required intermediate tool posture.

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