A Machine Tool Geometric Error Modeling Method Based on Dual Quaternions

Abstract

The present invention belongs to the technical field related to numerical control machining error analysis, and discloses a method for modeling machine tool geometric errors based on dual quaternions. The method includes the following steps: S1 constructing a machine tool coordinate system, a workpiece coordinate system, and a tool coordinate system of the machine tool; constructing the screw coordinates of each moving axis of the machine tool in the Plücker coordinate system, constructing a relational expression for converting screw parameters into dual quaternions, and converting the screw coordinates of each moving axis into dual quaternions; S2 using dual quaternion multiplication and screw kinematics modeling method to construct a dual quaternion kinematic model of the machine tool, converting the screw coordinates of various errors of the machine tool into dual quaternions, and adding the dual quaternions of various errors of the machine tool to the kinematic model, so as to establish a complete machine tool error model. Through the present invention, the problems of poor completeness, continuity, and minimality in error modeling in the prior art are solved.

Description

Technical Field

[0001] The present invention belongs to the technical field related to the analysis of numerical control machining errors, and more specifically, relates to a method for modeling machine tool geometric errors based on dual quaternions. Background Art

[0002] The basic connotation of machine tool geometric error modeling is to construct the actual position and attitude of the tool in the workpiece coordinate system of the machine tool under the influence of geometric errors. It is an important method for studying the accuracy of machine tools and is the theoretical basis for machine tool accuracy design, error compensation, and improvement of machining accuracy. The existing machine tool geometric error modeling methods mainly focus on the multi-body system theory and homogeneous coordinate transformation, coupling errors with the kinematic model, and constructing the error model of multi-axis machine tools through the homogeneous coordinate transformation method.

[0003] In addition, the exponential product of screws and the DH matrix are also used for the error modeling of multi-axis machine tools. However, due to the influence of computational complexity, completeness, continuity, and minimality of error modeling, the popularization of these methods is limited. Therefore, the established machine tool geometric error model needs to be simple and complete, and be able to accurately describe the actual position and attitude of the tool in the workpiece coordinate system under the action of errors. Summary of the Invention

[0004] Aiming at the above defects or improvement requirements of the existing technology, the present invention provides a method for modeling machine tool geometric errors based on the dual quaternion theory, which solves the problems of poor completeness, continuity, and minimality in error modeling in the existing technology.

[0005] To achieve the above object, according to the present invention, there is provided a method for modeling machine tool geometric errors based on dual quaternions, the method comprising the following steps:

[0006] S1 Construct the machine tool coordinate system, workpiece coordinate system, and tool coordinate system of the machine tool; construct the screw coordinates of each moving axis of the machine tool in the Plücker coordinate system, construct the relationship for converting screw parameters into dual quaternions, and convert the screw coordinates of each moving axis into dual quaternions;

[0007] S2 Use dual quaternion multiplication and screw kinematic modeling methods to construct the dual quaternion kinematic model of the machine tool, convert the screw coordinates of various errors of the machine tool into dual quaternions, and add the dual quaternions of various errors of the machine tool to the kinematic model to establish a complete machine tool error model.

[0008] Further preferably, in step S1, the machine tool is a four-axis machine tool.

[0009] Further preferably, in step S1, the relationship for converting screw parameters into dual quaternions is as follows:

[0010]

[0011] Among them, ω r is the real part of the quaternion in the real part of the dual quaternion, υ r is the imaginary part of the quaternion in the real part of the dual quaternion, ω d is the real part of the quaternion in the dual part of the dual quaternion, v d is the imaginary part of the quaternion in the dual part of the dual quaternion. θ represents the rotation angle around the space axis, and d represents the translation amount along the space axis. represents the direction vector of the space axis, represents the moment of the space axis.

[0012] Further preferably, in step S1, the screw coordinates of each motion axis are converted into dual quaternions according to the following relational expressions:

[0013]

[0014] Among them, represents the dual quaternion of the C-axis of the machine tool, represents the dual quaternion of the X-axis of the machine tool, represents the dual quaternion of the Z-axis of the machine tool, represents the dual quaternion of the B-axis of the machine tool, represents the dual quaternion of the tool pose in the workpiece coordinate system of the machine tool in the initial state, ω cr is the real part of the quaternion in the real part of the C-axis dual quaternion, v cr is the imaginary part of the quaternion in the real part of the C-axis dual quaternion, ω cd is the real part of the quaternion in the dual part of the C-axis dual quaternion, v cd is the imaginary part of the quaternion in the dual part of the C-axis dual quaternion, ε is the even symbol of the dual quaternion, ω xr is the real part of the quaternion in the real part of the X-axis dual quaternion, v xr is the imaginary part of the quaternion in the real part of the X-axis dual quaternion, ω xd is the imaginary part of the quaternion in the dual part of the X-axis dual quaternion, v xd is the imaginary part of the quaternion in the dual part of the X-axis dual quaternion, ω zr is the real part of the quaternion in the real part of the Z-axis dual quaternion, v zr is the imaginary part of the quaternion in the real part of the Z-axis dual quaternion, ω zd is the real part of the quaternion in the dual part of the Z-axis dual quaternion, v zd is the imaginary part of the quaternion in the dual part of the Z-axis dual quaternion, ω br is the real part of the quaternion in the real part of the B-axis dual quaternion, v br is the imaginary part of the quaternion in the real part of the B-axis dual quaternion, ω bdis the real part of the quaternion in the dual part of the B-axis dual quaternion, v bd is the imaginary part of the quaternion in the dual part of the B-axis dual quaternion, ω str is the real part of the quaternion in the real part of the tool pose dual quaternion in the machine tool workpiece coordinate system in the initial state, v str is the imaginary part of the quaternion in the real part of the tool pose dual quaternion in the machine tool workpiece coordinate system in the initial state, ω std is the real part of the quaternion in the dual part of the tool pose dual quaternion in the machine tool workpiece coordinate system in the initial state, v std is the imaginary part of the quaternion in the dual part of the tool pose dual quaternion in the machine tool workpiece coordinate system in the initial state. Further preferably, the representation form of the tool coordinate system dual quaternion in the workpiece coordinate system under the initial attitude is carried out according to the following relational expressions:

[0015]

[0016] where, x st , y st , z st are the position coordinates of the origin of the tool coordinate system in the workpiece coordinate system in the initial state, where the subscript st represents the initial state; i, j, and k represent the imaginary units in the quaternion, ε represents the dual quaternion even number symbol, ε≠0, ε n =0.

[0017] Further preferably, in step S2, the kinematic model of the machine tool is carried out according to the following relational expressions:

[0018]

[0019] where, where represents the tool pose dual quaternion in the workpiece coordinate system, represents the dual quaternion of the C-axis of the machine tool, represents the dual quaternion of the X-axis of the machine tool, represents the dual quaternion of the Z-axis of the machine tool, represents the dual quaternion of the B-axis of the machine tool, represents the tool pose dual quaternion in the machine tool workpiece coordinate system in the initial state.

[0020] Further preferably, in step S2, various errors of the machine tool include geometric errors related to position and geometric errors unrelated to position.

[0021] Further preferably, for the geometric errors unrelated to position, the position in the kinematic modeling process is before the motion axis motion dual quaternion.

[0022] Further preferably, for the position-related geometric errors, the specific position in the kinematic error model is after the kinematic axis motion dual quaternion, and in the six-degree-of-freedom geometric errors related to position, the linear geometric error dual quaternion is in the front and the angular geometric error dual quaternion is in the back.

[0023] Further preferably, in step S2, the complete machine tool error model is as follows:

[0024]

[0025] In the formula, pi in the subscript represents the dual quaternion form of the geometric error independent of position, pd in the subscript represents the dual quaternion form of the geometric error related to position, a in the subscript represents the angular error in the geometric error, and l in the subscript represents the linear error in the geometric error. is the dual quaternion of the C axis of the machine tool. is the C-axis linear geometric error in the form of a dual quaternion related to position. is the C-axis angular geometric error in the form of a dual quaternion related to position. is the geometric error of the X axis in the form of a dual quaternion independent of position. is the dual quaternion of the X axis of the machine tool. is the X-axis linear geometric error in the form of a dual quaternion related to position. is the X-axis angular geometric error in the form of a dual quaternion related to position. is the geometric error of the Z axis in the form of a dual quaternion independent of position. is the dual quaternion of the Z axis of the machine tool. is the Z-axis linear geometric error in the form of a dual quaternion related to position. is the Z-axis angular geometric error in the form of a dual quaternion related to position. is the geometric error of the B axis in the form of a dual quaternion independent of position. The dual quaternion of the B axis of the machine tool. is the B-axis linear geometric error in the form of a dual quaternion related to position. is the B-axis angular geometric error in the form of a dual quaternion related to position. is the representation form of the dual quaternion of the tool coordinate system in the workpiece coordinate system under the initial attitude.

[0026] Generally speaking, compared with the prior art, the above technical solution conceived by the present invention has the following beneficial effects:

[0027] 1. The machine tool error modeling method based on dual quaternion proposed by the present invention has higher completeness and simplicity. Each moving rigid body in the machine tool is represented in the form of a dual quaternion with 8 parameters. A kinematic model is constructed using dual quaternion multiplication. At the same time, geometric errors are represented in the form of dual quaternions and added to the constructed machine tool kinematic model in a specific order. This method inherits the advantages of both dual quaternions and screws in kinematic modeling. Compared with homogeneous coordinate transformation, this method reduces the redundancy of parameters and the computational cost.

[0028] 2. Compared with the screw error modeling, the method provided by the present invention does not involve exponential calculation, reduces the computational complexity. At the same time, this method has the completeness and accuracy of machine tool error modeling, can take all geometric errors into account, and the calculation results are accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 is a flowchart of the machine tool geometric error modeling method based on dual quaternion theory constructed according to the preferred embodiment of the present invention;

[0030] Figure 2 are the position-related geometric errors constructed according to the preferred embodiment of the present invention; among them, (a) the position-related error of the X-axis of the machine tool, (b) the position-related error of the Z-axis of the machine tool, (c) the position-related error of the C-axis of the machine tool, (d) the position-related error of the B-axis of the machine tool;

[0031] Figure 3 are the position-independent geometric errors constructed according to the preferred embodiment of the present invention; (a) the position-independent error of the X-axis of the machine tool, (b) the position-independent error of the Z-axis of the machine tool, (c) the position-independent error of the C-axis of the machine tool, (d) the position-independent error of the B-axis of the machine tool. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0032] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0033] As Figure 1 shown, a machine tool geometric error modeling method based on dual quaternion theory, the method includes the following steps:

[0034] Step 1: Establish a global coordinate system O at an arbitrary point on the machine tool, establish a workpiece coordinate system W on the motion chain where the workpiece is installed on the machine tool, and establish a tool coordinate system T at the center of the tool arc.

[0035] Step 2: Construct the screw coordinates of each motion axis. The Plücker coordinate system is used to create screw coordinates, which is a basic method for representing lines. Using screw coordinates, the dual quaternion can be rewritten in a more elegant form. The definition of the Plücker coordinate system includes:

[0036] P is any point on the given line;

[0037] is the direction vector of the line;

[0038] is the moment of the line;

[0039] is the six-dimensional Plücker coordinate;

[0040] A set of 8 equivalent screw coordinates can be converted into 8 dual quaternion parameters. The definitions of the parameters are as follows.

[0041]

[0042] In the formula, d represents the displacement along the axis, and θ represents the rotation angle around the axis. The conversion relationship from screw parameters to dual quaternions is as follows.

[0043]

[0044] Using the above conversion relationship, the dual quaternion expression forms of the screw coordinates of each motion axis of the machine tool can be obtained. The screw parameters of each motion axis of the four-axis ultra-precision machine tool in the base coordinate system are as follows.

[0045]

[0046] Using the above formula to convert the screw parameters into dual quaternions, as follows.

[0047]

[0048] In the formula represents the dual quaternion representation form of the tool coordinate system in the workpiece coordinate system (O) in the initial posture, which can be obtained by converting the screw parameters into dual quaternions. The specific form is as follows.

[0049]

[0050] In the formula, x st , y st , z st are the position coordinates of the origin of the tool coordinate system in the workpiece coordinate system. Since the tool coordinate system in the initial posture is in the same direction as the base coordinate system, there is no rotation amount.

[0051] Step 3: Construct the kinematic model of the four-axis ultra-precision machine tool according to the coordinate system transformation relationship as follows.

[0052]

[0053] In the formula is the pose of the tool coordinate system in the workpiece coordinate system under different motion parameters. To obtain this pose information intuitively, the dual quaternion is converted into the screw parameter form, specifically including the Plücker coordinates of the axis of the combined motion, the translation parameter along the axis, and the rotation parameter around the axis, as follows.

[0054]

[0055] Using Rodrigues' formula to convert the screw parameters into the homogeneous matrix form, the pose of the tool coordinate system in the workpiece coordinate system can be obtained as follows.

[0056]

[0057] In the formula, (n x , n y , n z ) represents the direction vector of the X-axis of the tool coordinate system in the workpiece coordinate system, (o x , o y , o z ) represents the direction vector of the Y-axis of the tool coordinate system in the workpiece coordinate system, (a x , a y , a z ) represents the direction vector of the Z-axis of the tool coordinate system in the workpiece coordinate system, (p x , p y , p z ) represents the position of the origin of the tool coordinate system in the workpiece coordinate system.

[0058] Step 4: Express the geometric errors of the machine tool in the form of screw coordinates, including geometric errors related to position and geometric errors independent of position. Among them, as shown in Table 2 and Figure 3 shown, the geometric meaning of the geometric errors independent of position is reflected in the multiplication order of the dual quaternions in kinematic modeling as: rotation around the X-axis → rotation around the Y-axis → rotation around the Z-axis. As shown in Table 1 and Figure 2 shown, the geometric meaning of the geometric errors related to position is reflected in the kinematic modeling order as: translation along the X-axis → translation along the Y-axis → translation along the Z-axis → rotation around the X-axis → rotation around the Y-axis → rotation around the Z-axis. Taking the error δ xx as an example, its screw parameter form is as follows, and it is converted into the dual quaternion form as follows.

[0059]

[0060]

[0061] Table 1 Position-related geometric error terms of a four-axis ultra-precision machine tool

[0062]

[0063] Table 2 Position-independent error terms of a four-axis ultra-precision machine tool

[0064]

[0065] Step 5: Add the error dual quaternion to the quaternion-based kinematic model in Step 3. The specific addition method is as follows: For the geometric error terms independent of position, since such errors determine the actual direction of the moving axis in the base coordinate system, the position of such errors in the multiplication order of kinematic modeling is before the moving axis motion dual quaternion. It should be noted that such errors cannot be directly superimposed with the corresponding position-related errors and placed after the moving axis motion dual quaternion, which will lead to theoretical errors in the kinematic error model. The specific position of the position-related geometric errors in the multiplication order of the kinematic error model is after the moving axis motion dual quaternion. Among the six-degree-of-freedom geometric errors related to position, the linear geometric error dual quaternion is in front and the angular geometric error dual quaternion is behind.

[0066] Finally, the error model of the ultra-precision four-axis machine tool under the action of errors is as follows.

[0067]

[0068] In the formula, the subscript pi represents the dual quaternion form of the geometric error independent of position, and the subscript pd represents the dual quaternion form of the geometric error related to position. The subscript a represents the angular error in the geometric error, and the subscript l represents the linear error in the geometric error.

[0069] Step 6: Convert the obtained dual quaternion under the influence of errors into screw coordinates, and use the Rodriguez formula to convert the screw parameters into homogeneous matrix form, then the direction vector of the X-axis of the tool coordinate system in the workpiece coordinate system, the direction vector of the Y-axis of the tool coordinate system in the workpiece coordinate system, the direction vector of the Z-axis of the tool coordinate system in the workpiece coordinate system, and the position of the origin of the tool coordinate system in the workpiece coordinate system can be obtained.

[0070] The above embodiments mainly explain the method for modeling the geometric errors of machine tools, without limiting the machine tool configuration. Similar three-axis machine tools, five-axis machine tools, etc. can all be applicable objects of this method.

[0071] Those skilled in the art can easily understand that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention should be included within the protection scope of the present invention.

Claims

1. A machine tool geometric error modeling method based on dual quaternions, characterized in that, The method includes the following steps: S1 Construct the machine tool coordinate system, workpiece coordinate system and tool coordinate system of the machine tool; construct the screw coordinates of each motion axis of the machine tool in the Plücker coordinate system, construct the relationship for converting the screw parameters into dual quaternions, and convert the screw coordinates of the respective motion axes into dual quaternions; S2 Use dual quaternion multiplication and screw kinematic modeling method to construct the dual quaternion kinematic model of the machine tool, convert the screw coordinates of various errors of the machine tool into dual quaternions, and add the dual quaternions of the various errors of the machine tool to the kinematic model, so as to establish a complete machine tool error model; In step S1, the machine tool is a four-axis machine tool; In step S1, the relationship for converting the screw parameters into dual quaternions is as follows: where ω r is the real part of the quaternion in the real part of the dual quaternion, υ r is the imaginary part of the quaternion in the real part of the dual quaternion, ω d is the real part of the quaternion in the dual part of the dual quaternion, v d is the imaginary part of the quaternion in the dual part of the dual quaternion, θ represents the rotation angle about the space axis, d represents the translation along the space axis, represents the direction vector of the space axis, represents the moment of the space axis.

2. The machine tool geometric error modeling method based on dual quaternions according to claim 1, characterized in that, In step S1, the conversion of the screw coordinates of the respective motion axes into dual quaternions is carried out according to the following relationship: Among them, represents the dual quaternion of the C-axis of the machine tool, represents the dual quaternion of the X-axis of the machine tool, represents the dual quaternion of the Z-axis of the machine tool, represents the dual quaternion of the B-axis of the machine tool, represents the dual quaternion of the tool pose in the workpiece coordinate system of the machine tool in the initial state, ω cr is the real part of the quaternion in the real part of the C-axis dual quaternion, v cr is the imaginary part of the quaternion in the real part of the C-axis dual quaternion, ω cd is the real part of the quaternion in the dual part of the C-axis dual quaternion, v cd is the imaginary part of the quaternion in the dual part of the C-axis dual quaternion, ε is the even symbol of the dual quaternion, ω xr is the real part of the quaternion in the real part of the X-axis dual quaternion, v xr is the imaginary part of the quaternion in the real part of the X-axis dual quaternion, ω xd is the real part of the quaternion in the dual part of the X-axis dual quaternion, v xd is the imaginary part of the quaternion in the dual part of the X-axis dual quaternion, ω zr is the real part of the quaternion in the real part of the Z-axis dual quaternion, v zr is the imaginary part of the quaternion in the real part of the Z-axis dual quaternion, ω zd is the real part of the quaternion in the dual part of the Z-axis dual quaternion, v zd is the imaginary part of the quaternion in the dual part of the Z-axis dual quaternion, ω br is the real part of the quaternion in the real part of the B-axis dual quaternion, v br is the imaginary part of the quaternion in the real part of the B-axis dual quaternion, ω bd is the real part of the quaternion in the dual part of the B-axis dual quaternion, v bd is the imaginary part of the quaternion in the dual part of the B-axis dual quaternion, ω str is the real part of the quaternion in the real part of the dual quaternion of the tool pose in the workpiece coordinate system of the machine tool in the initial state, v str is the imaginary part of the quaternion in the real part of the dual quaternion of the tool pose in the workpiece coordinate system of the machine tool in the initial state, ω std is the real part of the quaternion in the dual part of the dual quaternion of the tool pose in the workpiece coordinate system of the machine tool in the initial state, v std is the imaginary part of the quaternion in the dual part of the dual quaternion of the tool pose in the workpiece coordinate system of the machine tool in the initial state.

3. The machine tool geometric error modeling method based on dual quaternions according to claim 2, characterized in that, The representation form of the dual quaternion of the tool coordinate system in the workpiece coordinate system under the initial state It is carried out according to the following relational expressions: where x st , y st , z st are the position coordinates of the origin of the tool coordinate system on the X, Y, and Z axes in the workpiece coordinate system in the initial state, where the subscript st represents the initial state; i, j, and k are the imaginary units in the quaternion, ε is the dual quaternion dual number symbol, ε≠0, ε n = 0.

4. The machine tool geometric error modeling method based on dual quaternions according to claim 1, characterized in that, In step S2, the kinematic model is as follows: Among them, is the dual quaternion of the tool pose in the workpiece coordinate system, is the dual quaternion of the C-axis of the machine tool, is the dual quaternion of the X-axis of the machine tool, is the dual quaternion of the Z-axis of the machine tool, represents the dual quaternion of the B-axis of the machine tool, is the dual quaternion of the tool pose in the workpiece coordinate system of the machine tool under the initial state.

5. The machine tool geometric error modeling method based on dual quaternions according to claim 1, characterized in that, In step S2, the various errors of the machine tool include geometric errors related to position and geometric errors unrelated to position.

6. The machine tool geometric error modeling method based on dual quaternions according to claim 5, characterized in that, For the geometric errors unrelated to position, the position in the kinematic modeling process is before the motion axis motion dual quaternion.

7. The machine tool geometric error modeling method based on dual quaternions according to claim 6, characterized in that, For the geometric errors related to position, the specific position in the kinematic error model is after the motion axis motion dual quaternion, and among the six-degree-of-freedom geometric errors related to position, the linear geometric error dual quaternion is in front and the angular geometric error dual quaternion is behind.

8. The machine tool geometric error modeling method based on dual quaternions according to claim 7, characterized in that, In step S2, the complete machine tool error model is as follows: In the formula, the subscript \(p_i\) represents the dual quaternion form of the geometric error independent of position, the subscript \(p_d\) represents the dual quaternion form of the geometric error related to position, the subscript \(a\) represents the angular error in the geometric error, and the subscript \(l\) represents the linear error in the geometric error. is the dual quaternion of the C-axis of the machine tool. is the C-axis linear geometric error in the dual quaternion form related to position. is the C-axis angular geometric error in the dual quaternion form related to position. is the X-axis geometric error in the dual quaternion form independent of position. is the dual quaternion of the X-axis of the machine tool. is the X-axis linear geometric error in the dual quaternion form related to position. is the X-axis angular geometric error in the dual quaternion form related to position. is the Z-axis geometric error in the dual quaternion form independent of position. is the dual quaternion of the Z-axis of the machine tool. is the Z-axis linear geometric error in the dual quaternion form related to position. is the Z-axis angular geometric error in the dual quaternion form related to position. is the B-axis geometric error in the dual quaternion form independent of position. The dual quaternion of the B-axis of the machine tool. is the B-axis linear geometric error in the dual quaternion form related to position. is the B-axis angular geometric error in the dual quaternion form related to position. is the dual quaternion representation form of the tool coordinate system in the workpiece coordinate system under the initial state.

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