An online estimation and compensation method for contour error in five-axis turning and milling machining

Abstract

A method for online estimation and compensation of contour error in five-axis milling and turning machining comprises the following steps: (1) constructing and solving the kinematic chain model of a five-axis milling and turning CNC machine tool based on the screw theory, and deriving the forward and inverse kinematic solutions of the tool tip point and the tool axis direction; (2) estimating the tracking error of each active drive, and obtaining the estimated contour error value at the next moment. The estimated axis component of the contour error is fed back to the position command of each closed-loop servo drive with proportional gain; (3) substituting the predicted compensation amount of the contour error of each axis into the ideal motion position of the corresponding feed axis to obtain the tool position after compensation. The tool position trajectory data of the part to be machined is converted into NC code that can be recognized by the five-axis milling and turning CNC machine tool through a post-processor, and the coordinates of each feed axis in the original NC code are replaced for actual machining. This method effectively improves the contour accuracy of five-axis milling and turning machining of complex curved surface parts, and the calculation process is stable.

Description

Technical Field

[0001] The invention relates to a method for estimating and compensating contour errors when five-axis turning and milling composite machining is performed on mechanical parts, and belongs to the technical field of dynamic error compensation of numerically controlled machine tools. Background Art

[0002] Multi-axis milling-turning technology can complete all turning, milling, drilling, boring, tapping and other processing in one clamping. Its wide range of processes and strong capabilities have made it one of the most advanced mechanical processing technologies in equipment manufacturing technology. The contour error and compensation technology for two-axis and three-axis linkage machine tools has been relatively complete. Since the five-axis contour error includes not only the tool tip contour error, but also the tool orientation contour error, and the two are coupled with each other, the estimation and compensation of the five-axis contour error is more difficult than the three-axis contour control. In particular, the current research on the contour error estimation and compensation technology of multi-axis milling-turning is relatively small and difficult. Therefore, the study of the online estimation and compensation method of the contour error of the five-axis milling-turning compound machining is of great significance to improve the machining accuracy of the five-axis milling-turning compound and improve the performance of the five-axis milling-turning compound machining center.

[0003] In terms of improving machining accuracy, five-axis milling and turning mainly processes some complex curved surface parts, and related research mainly focuses on the spatial error control of the machine tool itself, but there is less research on the precision control of the trajectory itself when machining complex curved surface parts. The published technical literature "Spatial Error Modeling and Compensation of Milling and Turning CNC Machine Tools" (Machine Tools and Hydraulics, 2023, 51(24): 157-163) realizes the offline compensation of five-axis milling and turning machining errors, improves the flexible machining and manufacturing of machine tools and flexible error compensation, and solves the limitations of current methods, but does not consider how to improve the precision of the machining trajectory itself of complex curved surface parts.

[0004] Chinese patent document CN116859821A discloses a post-processing method for optimizing the trajectory of a four-axis turning-milling compound machining. Taking the angle between adjacent tool position points as a constraint, a tool position point sparse processing algorithm is constructed to thin out the tool position trajectory after the tool position trajectory optimization algorithm is optimized. This method can effectively improve the machining accuracy, but this method only compensates for the contour error that has been generated, and cannot perform online real-time estimation and compensation for the error, and cannot optimize the trajectory optimization of a five-axis turning-milling compound machining with a B-swing head. Summary of the invention

[0005] The present invention aims to overcome the defects of the prior art and provide a method for online estimation and compensation of contour errors in five-axis turning and milling to improve the contour accuracy of five-axis turning and milling for machining complex curved surface parts.

[0006] The five-axis turning-milling composite machining contour error online estimation and compensation method of the present invention comprises the following steps:

[0007] (1) Based on the screw theory, the kinematic chain model of the five-axis turning and milling compound CNC machine tool is constructed and solved, and then the forward and inverse kinematic solutions of the tool tip point and tool axis direction are derived to achieve synchronous linear and rotational motion;

[0008] (2) According to the inverse Jacobian matrix, the tracking error of each active drive is estimated by interpolating the position command to obtain the estimated value of the contour error at the next moment. The estimated axis component of the contour error is fed back to the position command of each closed-loop servo drive with a proportional gain;

[0009] (3) The predicted compensation amount of each axis contour error is substituted into the ideal motion position of the corresponding feed axis to obtain the compensated tool position. The tool trajectory data of the part to be processed is converted into NC code that can be recognized by the five-axis turning and milling compound CNC machine tool through the post-processor, replacing the coordinates of each feed axis in the original NC code, and then used for actual processing, thereby improving the contour accuracy of the tool processing trajectory and ultimately improving the contour accuracy of complex curved surface parts.

[0010] The step (1) specifically comprises:

[0011] ① Establish the machine tool coordinate system, workpiece coordinate system and tool coordinate system;

[0012] ② Establish the direction vectors of each moving part of the five-axis turning and milling compound CNC machine tool;

[0013] ③Establish a kinematic chain model from tool to workpiece;

[0014] ④Construct the forward kinematics model of the five-axis turning-milling compound CNC machine tool based on the screw theory;

[0015] ⑤ Derive the inverse kinematics solution of the moving direction of the tool tip and the inverse kinematics solution of the rotation direction of the tool axis.

[0016] The process of establishing the direction vectors of the moving parts of the five-axis turning and milling compound CNC machine tool is as follows:

[0017] The setting of the five-axis turning and milling compound CNC machine tool is as follows: the direction of the tool is changed by the rotation of the swing head B, the position of the tool is changed by the movement of X, Y, and Z, the C axis drives the workpiece to rotate, and the workpiece coordinate system is consistent with the machine tool coordinate system;

[0018] Machine tool coordinate system (X b ,Y b ,Z b ), tool coordinate system (X t ,Y t ,Z t ), workpiece coordinate system (X w ,Y w ,Z w ), let P = (Px ,P y ,P z ) T Indicates the position of the tool tip, O = (O i ,O j ,O k ) T represents the tool axis vector, (v x ,v y ,v z ) and (ω b ,ω c ) represent the unit vectors in the positive directions of the moving axis and the rotating axis, respectively, q b and q c are points on the corresponding rotation axes.

[0019] The process of establishing the kinematic chain model from tool to workpiece in ③ is:

[0020] Using the mathematical modeling theory of robots, the relative position and direction of the nth joint relative to the base coordinate system is expressed by a 4×4 homogeneous transformation matrix:

[0021]

[0022] According to the screw theory, for a revolute joint:

[0023]

[0024] where ω i is the unit vector of the revolute joint, q i is a point on the rotation axis; for a five-axis milling CNC machine tool, the unit vector of the rotation joint on the B axis and the C axis is represented by ω b =[0 1 0] T and ω c =[0 0 1] T ;

[0025] For moving joints:

[0026]

[0027] The homogeneous transformation matrix of the workpiece chain is:

[0028]

[0029] Similarly, the homogeneous transformation matrix of the tool chain is obtained:

[0030]

[0031] Since the workpiece chain and tool chain are represented analytically, the homogeneous transformation matrix of the workpiece chain and the homogeneous transformation matrix of the tool chain (the above two formulas) are combined to obtain the full motion chain model of the five-axis turning and milling compound CNC machine tool:

[0032]

[0033] The process of constructing the positive kinematics model of the five-axis turning-milling compound CNC machine tool based on the screw theory is as follows:

[0034] Assume that the position vector and direction vector of the tool relative to the tool coordinate system are r pt and r ot , then the positive kinematics of the five-axis turning-milling compound CNC machine tool is expressed as:

[0035]

[0036] Here P and O represent the position and direction of the tool relative to the workpiece; after solving the forward kinematics, the inverse kinematics model is established to find (θ c , x, y, z, θ b ) is the reference position command; the terms with moving joints are removed from the equation, further yielding the forward kinematic solution for the direction vector:

[0037]

[0038] The process of deriving the inverse kinematics solution of the moving direction of the tool tip and the inverse kinematics solution of the rotating direction of the tool axis in ⑤ is:

[0039] For the inverse kinematic solution, let u = r ot , v = O, the unit vectors of the B axis and the C axis and their vector products are linearly independent, so a new variable z is defined as:

[0040] z=k1ω c +k2ω b +k3(ω c ×ω b );

[0041] Solve the rotation angles of the B and C axes of the five-axis turning and milling compound CNC machine tool under given CL data:

[0042]

[0043] To solve the inverse kinematics of the moving joint, simplify the formula:

[0044]

[0045] For commands that move axes:

[0046] S p =[Sx S y S z ] T ;

[0047] Then rewrite

[0048] Finally, the inverse kinematic solution of translational motion is obtained:

[0049]

[0050] The derivation process of the inverse Jacobian matrix in step (2) is:

[0051] According to the screw theory, the instantaneous spatial velocity of the tool relative to the workpiece is:

[0052]

[0053] In the workpiece coordinate system, the position of the tool tip is:

[0054]

[0055] Will Bring in Get the Jacobian matrix:

[0056]

[0057] Derive the Jacobian matrix of the five-axis turning-milling CNC machine tool:

[0058]

[0059] The specific process of estimating the tracking error of each active drive by interpolating the position command in step (2) to obtain the estimated value of the contour error at the next moment is:

[0060] The general expression of the z-domain transfer function G(z) of the CNC feed system is:

[0061]

[0062] Where n is the system order, a1, a2, …, a n and b1,b2,…,b n is the system parameter, based on which the estimated value of the actual motion position of the physical axis in the next sampling period is obtained:

[0063]

[0064] According to the known forward and inverse kinematic solutions of each transmission axis, the actual tool position vector estimate for the next sampling period is calculated:

[0065]

[0066] The contour error vector of a sampling period is written as in Indicates the contour error of the tool tip point, Indicates the direction error of the tool axis. The predicted compensation amount of the contour error of each physical axis is calculated according to the Jacobian matrix of the five-axis turning and milling compound CNC machine tool:

[0067]

[0068] The method of the present invention constructs and solves the motion chain model of a five-axis turning-milling compound CNC machine tool based on the screw theory, derives the forward and inverse kinematic solutions of the tool tip point and the tool axis direction, estimates the tracking errors of each active drive by interpolating position commands, calculates the tool machining trajectory contour error compensation value by using the components of the contour error vector in each machining feed axis, generates compensated CNC machining codes, and thus effectively improves the machining contour accuracy of the five-axis turning-milling compound for complex curved surface parts.

[0069] The present invention constructs and solves the motion chain model of the five-axis turning-milling compound CNC machine tool based on the screw theory. In view of the advantages of the screw theory itself, the number of matrix multiplications of the system kinematic solution is reduced and will not be affected by numerical pathology or singularities, so that the calculation amount of the present invention is small and the system has strong anti-interference ability, and can realize the synchronous and accurate calculation of the tool tip point and tool axis direction errors; the tracking error of each active drive is estimated online by interpolating the position command, and the contour error estimate at the next moment can be obtained, which can compensate for the contour error of the tool machining trajectory, and can significantly improve the contour accuracy of the five-axis turning-milling compound with a B-swing head for complex curved surface parts, and the calculation process is stable. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 It is a flow chart of the method for online estimation and compensation of contour error in five-axis turning and milling composite machining of the present invention.

[0071] Figure 2 It is a schematic diagram of the kinematic chain model of a five-axis turning-milling compound CNC machine tool; wherein, curve 1 represents the workpiece kinematic chain, curve 2 represents the complete kinematic chain, and curve 3 represents the tool kinematic chain.

[0072] Figure 3 It is a simplified model diagram of the kinematic chain of a five-axis turning-milling compound CNC machine tool.

[0073] Figure 4 This is a tool tip position contour error diagram obtained by the method of the present invention. Among them: B1 axis represents the processing time, the unit is s; B2 axis represents the tool tip position contour error value, the unit is mm; curve 1 represents the tool tip position contour error value; curve 2 represents the average error value of the tool tip position contour error.

[0074] Figure 5 This is a tool axis posture profile error diagram obtained by the method of the present invention. Among them: B1 axis represents the processing time, the unit is s; B2 axis represents the tool axis posture profile error value, the unit is rad, curve 1 represents the tool axis posture profile error value, and curve 2 represents the average error value of the tool axis posture profile error. DETAILED DESCRIPTION

[0075] Figure 1 A flow chart of the online estimation and compensation method for contour error in five-axis turning and milling machining of the present invention is given. Figure 2 Taking a horizontal five-axis turning-milling compound CNC machine tool as an example, the kinematic chain model of the present invention is described in detail, wherein curve 1 represents the workpiece kinematic chain, curve 2 represents the complete kinematic chain, and curve 3 represents the tool kinematic chain. Figure 3 A simplified model of the kinematic chain is given for a better understanding of the model.

[0076] The present invention discloses a method for online estimation and compensation of contour errors in five-axis turning and milling composite machining, see Figure 1 , the specific steps are as follows.

[0077] 1. Based on the screw theory, the kinematic chain model of the five-axis turning-milling compound CNC machine tool is constructed and solved, and then the forward and inverse kinematic solutions of the tool tip point and tool axis direction are derived to achieve synchronous linear and rotational motion.

[0078] Since the online estimation method of contour error in five-axis turning and milling needs to estimate the tracking error of each active drive by interpolating position commands, the forward and inverse kinematic solutions of the tool tip point and tool axis direction are a prerequisite, including:

[0079] Establish the machine tool coordinate system (X b ,Y b ,Z b ), workpiece coordinate system (X w ,Y w ,Z w ), tool coordinate system (X t ,Y t ,Z t );

[0080] Establish the direction vectors of each moving part of the five-axis milling CNC machine tool;

[0081] Establish a kinematic chain model from tool to workpiece;

[0082] Construct the forward kinematics model of five-axis turning-milling compound CNC machine tool based on screw theory;

[0083] The inverse kinematics solutions of the moving direction of the tool tip and the rotating direction of the tool axis are derived.

[0084] Taking a horizontal five-axis turning-milling compound CNC machine tool as an example, the settings are: the rotation center of the swing head B attached to the Z axis is parallel to the Y axis, the direction of the tool is changed by the rotation of the swing head B, the position of the tool is changed by the movement of X, Y, and Z, the C axis drives the workpiece to rotate, and the workpiece coordinate system is in the same direction as the machine tool coordinate system.

[0085] Let: P = (P x ,P y ,P z ) T , indicating the position of the tool tip; O = (O i ,O j ,O k ) T , represents the tool axis vector; (v x ,v y ,v z ) and (ω b ,ω c ) represent the unit vectors in the positive directions of the moving axis and the rotating axis, respectively, q b and q c are points on the corresponding rotation axes respectively. Using the mathematical modeling theory of robots, the relative position and direction of the nth joint relative to the base coordinate system is expressed by a 4×4 homogeneous transformation matrix:

[0086]

[0087] From the screw theory, we know that for a revolute joint:

[0088]

[0089] where ω i is the unit vector of the revolute joint, q i is a point on the rotation axis. For a five-axis milling CNC machine tool, the unit vector of the rotation joint on the B axis and the C axis is represented by ω b =[0 1 0] T and ω c =[0 0 1] T For moving joints:

[0090]

[0091] The homogeneous transformation matrix of the workpiece chain is:

[0092]

[0093] Similarly, the homogeneous transformation matrix of the tool chain can be obtained:

[0094]

[0095] Since the workpiece chain and tool chain are expressed analytically, the full kinematic chain model of the five-axis turning-milling compound CNC machine tool can be obtained by combining equation (4) and equation (5):

[0096]

[0097] Assume that the position vector and direction vector of the tool relative to the tool coordinate system are r pt and r ot , then the forward kinematics of the five-axis turning-milling compound CNC machine tool can be expressed as:

[0098]

[0099] Here P and O represent the position and direction of the tool relative to the workpiece. After solving the forward kinematics, the inverse kinematics model can be established to find (θ c , x, y, z, θ b ) is the reference position command. It is known from research that no matter what the position and order of each moving joint is, moving the joint will not change the direction vector. Therefore, the terms with moving joints are removed from the equation, and the forward kinematic solution of the direction vector is further obtained:

[0100]

[0101] For the inverse kinematic solution, let u = r ot , v = O, the unit vectors of the B axis and the C axis and their vector products are linearly independent, then a new variable z can be defined as:

[0102] z=k1ω c +k2ω b +k3(ω c ×ω b ) (9)

[0103] Solve the rotation angles of the B and C axes of the five-axis turning and milling compound CNC machine tool under given CL data:

[0104]

[0105] The inverse kinematics solution of the moving joint can be simplified to:

[0106]

[0107] For commands that move axes:

[0108] S p =[S x S y S z ] T (13)

[0109] Then, we can rewrite equation (12) and finally obtain the inverse kinematic solution of translational motion:

[0110]

[0111] 2. Through the established forward and inverse kinematic solutions of the tool tip point and tool axis direction, the inverse Jacobian matrix is ​​solved, and the tracking error of each active drive is estimated by interpolating the position command to obtain the estimated value of the contour error at the next moment. The estimated axis component of the contour error is fed back to the position command of each closed-loop servo drive with proportional gain.

[0112] According to the screw theory, the instantaneous spatial velocity of the tool relative to the workpiece is:

[0113]

[0114] In the workpiece coordinate system, the position of the tool tip is:

[0115]

[0116] Substituting equation (15) into equation (16) yields the Jacobian matrix:

[0117]

[0118] Derive the Jacobian matrix of the five-axis turning-milling CNC machine tool:

[0119]

[0120] The general expression of the z-domain transfer function G(z) of the CNC feed system is:

[0121]

[0122] Where n is the system order, a1, a2, …, a n and b1,b2,…,b n is the system parameter, based on which the estimated value of the actual motion position of the physical axis in the next sampling period is obtained:

[0123]

[0124] According to the known forward and inverse kinematic solutions of each transmission axis, the actual tool position vector estimate for the next sampling period is calculated:

[0125]

[0126] The contour error vector of a sampling period is written as in Indicates the contour error of the tool tip point, Indicates the direction error of the tool axis. The predicted compensation amount of the contour error of each physical axis is calculated according to the Jacobian matrix of the five-axis turning and milling compound CNC machine tool:

[0127]

[0128] 3. Substitute the predicted compensation amount of each axis contour error into the ideal motion position of the corresponding feed axis to obtain the compensated tool position. The tool trajectory data of the part to be processed is converted into NC code that can be recognized by the five-axis turning and milling compound CNC machine tool through the post-processor, replacing the coordinates of each feed axis in the original NC code, and then used for actual processing, thereby improving the contour accuracy of the tool processing trajectory and ultimately improving the contour accuracy of complex curved surface parts.

[0129] Figure 4 The figure shows the contour error diagram of the tool tip position obtained by the method of the present invention, wherein the B1 axis represents the machining time in seconds, the B2 axis represents the contour error value of the tool tip position in mm, the curve 1 represents the contour error value of the tool tip position obtained by the method of the present invention, and the curve 2 represents the average value of the contour error of the tool tip position obtained by the method of the present invention. It can be seen that the maximum contour error of the tool tip position obtained by the method of the present invention is about 11x10 -3 mm, the average value of the contour error of the tool tip position is 4.5x10 -3 mm or so.

[0130] Figure 5 The figure shows the tool axis posture profile error diagram obtained by the method of the present invention, wherein the B1 axis represents the processing time in seconds, the B2 axis represents the tool axis posture profile error value in rad, curve 1 represents the tool axis posture profile error value obtained by the method of the present invention, and curve 2 represents the tool axis posture profile error average value obtained by the method of the present invention; it can be seen that the maximum value of the tool axis posture profile error obtained by the method of the present invention is about 9.1x10 -3 rad, the average value of the tool axis posture contour error is 3.6x10 -3 rad or so.

[0131] It can be seen from the accompanying drawings that the method of the present invention can effectively reduce the contour error of the tool tip and tool axis posture during the five-axis turning and milling compound processing, and improve the contour tracking accuracy of the five-axis turning and milling compound processing. On the basis of realizing the synchronous and accurate calculation of the tool tip and tool axis posture errors, the calculation amount is small and the system has strong anti-interference ability, and can compensate for the contour error of the tool processing trajectory, which is of great significance to improving the contour accuracy of the five-axis turning and milling compound with B swing head for complex curved surface parts.

Claims

1. A method for online estimation and compensation of contour error in five-axis turning and milling machining, characterized in that: The following steps are involved: (1) Based on the screw theory, the kinematic chain model of the five-axis turning and milling CNC machine tool is constructed and solved, and then the forward and inverse kinematic solutions of the tool tip and tool axis direction are derived to achieve synchronous linear and rotational motion; (2) According to the inverse Jacobian matrix, the tracking error of each active drive is estimated by interpolating the position command to obtain the estimated value of the contour error at the next moment. The estimated axis component of the contour error is fed back to the position command of each closed-loop servo drive with a proportional gain; (3) Substitute the predicted compensation amount of each axis contour error into the ideal motion position of the corresponding feed axis to obtain the compensated tool position. The tool position trajectory data of the part to be processed is converted into NC code that can be recognized by the five-axis turning and milling compound CNC machine tool through the post-processor, replacing the coordinates of each feed axis in the original NC code, and then used for actual processing, thereby improving the contour accuracy of the tool processing trajectory and ultimately improving the contour accuracy of complex curved surface parts; The specific implementation process of step (1) includes: ① Establish the machine tool coordinate system, workpiece coordinate system and tool coordinate system; ② Establish the direction vectors of each moving part of the five-axis turning and milling compound CNC machine tool; ③Establish a kinematic chain model from tool to workpiece; ④Construct the forward kinematics model of the five-axis turning-milling compound CNC machine tool based on the screw theory; ⑤ Derive the inverse kinematics solution of the moving direction of the tool tip and the inverse kinematics solution of the rotation direction of the tool axis; The process of establishing the direction vectors of the moving parts of the five-axis turning and milling compound CNC machine tool is as follows: The setting of the five-axis turning and milling compound CNC machine tool is as follows: the direction of the tool is changed by the rotation of the swing head B, the position of the tool is changed by the movement of X, Y, and Z, the C axis drives the workpiece to rotate, and the workpiece coordinate system is consistent with the machine tool coordinate system; Machine tool coordinate system (X b ,Y b ,Z b ), tool coordinate system (X t ,Y t ,Z t ), workpiece coordinate system (X w ,Y w ,Z w ), let P = (P x ,P y ,P z ) T Indicates the position of the tool tip, O = (O i ,O j ,O k ) T represents the tool axis vector, (v x ,v y ,v z ) and (ω b ,ω c ) represent the unit vectors in the positive directions of the moving axis and the rotating axis, respectively, q b and q c are points on the corresponding rotation axes respectively; The process of establishing the kinematic chain model from tool to workpiece in ③ is: Using the mathematical modeling theory of robots, the relative position and direction of the nth joint relative to the base coordinate system is expressed by a 4×4 homogeneous transformation matrix: According to the screw theory, for a revolute joint: where ω i is the unit vector of the revolute joint, q i is a point on the rotation axis; for a five-axis milling CNC machine tool, the unit vector of the rotation joint on the B axis and the C axis is represented by ω b =[0 1 0] T and ω c =[0 0 1] T ; For moving joints: The homogeneous transformation matrix of the workpiece chain is: Similarly, the homogeneous transformation matrix of the tool chain is obtained: Since the workpiece chain and tool chain are represented analytically, the homogeneous transformation matrix of the workpiece chain and the homogeneous transformation matrix of the tool chain are combined to obtain the full motion chain model of the five-axis turning and milling compound CNC machine tool: The derivation process of the inverse Jacobian matrix in step (2) is: According to the screw theory, the instantaneous spatial velocity of the tool relative to the workpiece is: In the workpiece coordinate system, the position of the tool tip is: Will Bring in Get the Jacobian matrix: Derive the Jacobian matrix of the five-axis turning-milling CNC machine tool:

2. The method for online estimation and compensation of contour error in five-axis turning and milling machining according to claim 1 is characterized in that: The process of constructing the positive kinematics model of the five-axis turning-milling compound CNC machine tool based on the screw theory is as follows: Assume that the position vector and direction vector of the tool relative to the tool coordinate system are r pt and r ot , then the positive kinematics of the five-axis turning-milling compound CNC machine tool is expressed as: Here P and O represent the position and direction of the tool relative to the workpiece; after solving the forward kinematics, the inverse kinematics model is established to find (θ c , x, y, z, θ b ); the term with the moving joint is removed from the equation, resulting in the forward kinematic solution for the orientation vector:

3. The method for online estimation and compensation of contour error in five-axis turning and milling machining according to claim 1 is characterized in that: The process of deriving the inverse kinematics solution of the moving direction of the tool tip and the inverse kinematics solution of the rotating direction of the tool axis in ⑤ is: For the inverse kinematic solution, let u = r ot , v = O, the unit vectors of the B axis and the C axis and their vector products are linearly independent, so a new variable z is defined as: z=k1ω c +k2ω b +k3(ω c ×ω b ); Solve the rotation angles of the B and C axes of the five-axis turning and milling compound CNC machine tool under given CL data: To solve the inverse kinematics of the moving joint, simplify the formula: For commands that move axes: S p =[S x S y S z ] T ; Then rewrite Finally, the inverse kinematic solution of translational motion is obtained:

4. The method for online estimation and compensation of contour error in five-axis turning and milling machining according to claim 1 is characterized in that: The specific process of estimating the tracking error of each transmission shaft by interpolating the position command in step (2) to obtain the contour error estimate at the next moment is: The general expression of the z-domain transfer function G(z) of the CNC feed system is: Where n is the system order, a1, a2, ..., a n and b1, b2, …, b n is the system parameter, based on which the estimated value of the actual motion position of the physical axis in the next sampling period is obtained: According to the known forward and inverse kinematic solutions of each transmission axis, the actual tool position vector estimate for the next sampling period is calculated: The contour error vector of a sampling period is written as in Indicates the contour error of the tool tip point, Indicates the direction error of the tool axis. The predicted compensation amount of the contour error of each physical axis is calculated according to the Jacobian matrix of the five-axis turning and milling compound CNC machine tool:

Images