A polishing path determination method and a polishing method for a curved workpiece

Abstract

This invention discloses a method for determining the polishing path and a polishing method for curved workpieces. The determination method includes the following steps: determining an initial polishing path based on an Archimedes spiral; selecting any two adjacent initial path points in the initial polishing path; determining an error function based on the removal function of each of the two adjacent initial path points; and adjusting the optimized polishing spacing of each adjacent initial path point in the initial polishing path using Newton's downhill method, thereby determining the polishing path of the curved workpiece. This invention generates an optimized path by dynamically adjusting the spacing of the initial path points, breaking the periodic arrangement pattern of traditional spiral / grating paths, suppressing mid-frequency ripple errors caused by periodic motion, and replacing the random layout of pseudo-random paths with a deterministic optimization mechanism based on error convergence, avoiding the problem of local uneven material removal caused by high randomness, and reducing excessive dependence on the dynamic performance of ultra-high-performance machine tools.

Description

Technical Field

[0001] This invention belongs to the field of CNC machining technology, and particularly relates to a method for determining the polishing path and a polishing method for curved workpieces. Background Technology

[0002] Optical systems are widely used in key fields such as aerospace, defense, and biomedicine. With increasingly stringent requirements for imaging quality, the application of high-precision complex curved surfaces (such as aspherical and freeform surfaces) is increasing. Compared with traditional planar and spherical components, complex curved surfaces have significant advantages such as high imaging quality, large field of view, and low light energy loss. In order to meet the stringent surface accuracy requirements of high-precision complex curved surfaces (reducing roughness, controlling waviness, and improving shape accuracy), ultra-precision polishing has become an indispensable key step in manufacturing high-precision optical components.

[0003] Currently, there are two main types of polishing path control technologies. One type is the traditional path, which primarily uses regular, highly repeatable trajectories (such as grating or spiral paths). However, its inherent periodic motion introduces mid-frequency waviness errors, which scatter light and severely degrade the performance of the optical system. The other type is the pseudo-random polishing path, which generates seemingly random, highly non-repetitive trajectories through algorithms. Its core idea is to break the periodic motion, aiming to effectively suppress mid-frequency waviness errors caused by periodic motion. However, it places extremely high demands on the dynamic performance of the machine tool (acceleration, jerk, and trajectory tracking accuracy), making it difficult and costly to implement. Its high randomness may also lead to localized uneven material removal, which in turn impairs the control of global shape accuracy (such as surface PV value and RMS value).

[0004] Therefore, existing technologies struggle to effectively reduce mid-frequency waviness error while simultaneously achieving high shape accuracy control and avoiding excessive reliance on ultra-high-performance machine tools. Summary of the Invention

[0005] The purpose of this invention is to provide a solution that overcomes the problem that existing technologies struggle to effectively reduce mid-frequency waviness error while simultaneously achieving high shape accuracy control and avoiding excessive reliance on ultra-high-performance machine tools.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows: This invention discloses a method for determining the polishing path of a curved workpiece, comprising the following steps: S1. Establish a workpiece coordinate system with the center of the curved workpiece as the origin, and determine an initial polishing path based on the Archimedes spiral according to the workpiece coordinate system and the preset spacing parameters. The initial polishing path includes the position information of multiple initial path points and the initial spacing between adjacent initial path points. S2. Select any two adjacent initial path points and determine their respective removal functions, denoted as the first removal function and the second removal function, respectively. S3. Determine the error function based on the first removal function and the second removal function, and determine the adjustment amount in Newton's downhill method based on the error function and the initial distance between two adjacent initial path points; S4. Iteratively adjust the initial spacing between two adjacent initial path points in the initial polishing path according to the adjustment amount until the error function converges to the preset allowable error value, and obtain the optimized polishing spacing between the two adjacent initial path points. S5. Following the above method, determine the optimized polishing spacing of the remaining adjacent initial path points, and use multiple optimized polishing spacings to adjust the position information of each initial path point to obtain multiple optimized path points in the workpiece coordinate system. S6. Determine the polishing path of the curved workpiece based on the multiple optimized path points.

[0007] S3 specifically refers to: S31. Determine the overlap function based on the first removal function and the second removal function; S32. Determine the maximum value of the first removal function and the maximum value of the second removal function, and denot them as the first maximum value and the second maximum value, respectively; S33. Determine the error function based on any two of the first maximum value, the second maximum value, and the overlap function.

[0008] Preferably, S33 is as follows: Determine the maximum value of the overlap function, denoted as the third maximum value; The error function is determined by the mean square error of any two of the first, second, and third maximum values.

[0009] Preferably, S6 is as follows: S61. Based on the Euclidean distance between adjacent optimized path points, the following method is adopted: Hartley-Judd The method calculates the nodal vectors of the B-spline interpolation curve to obtain the non-uniformly distributed nodal values; S62. Construct an interpolation curve equation based on the node values ​​to generate a continuous optimized polishing path, and use the optimized polishing path as the polishing path of the curved workpiece.

[0010] This invention also discloses a polishing method for curved workpieces, applied to the polishing path determined by the polishing path determination method for curved workpieces. The polishing method includes the following steps: A tool coordinate system coinciding with the workpiece coordinate system is established with the starting point of the initial polishing path as the origin. The optimized polishing path is discretized to obtain multiple tool point position information for polishing curved workpieces in the tool coordinate system; Based on the tool point position information and the preset precession angle, determine the tool point attitude information corresponding to each tool point position information in the workpiece coordinate system; Polishing of curved workpieces is performed based on multiple tool point posture information.

[0011] Preferably, the optimized polishing path is discretized to obtain multiple tool point position information for polishing the curved workpiece in the tool coordinate system, specifically: Determine the local radius curvature at the current contact point between the workpiece and the tool based on the surface equation:

[0012]

[0013] In the formula, R i The local radius of curvature at the current contact point; x and y These are the x and y coordinates of the current contact point in the tool coordinate system, with the first current contact point being the starting point of the optimized polishing path. z This is the definition of the surface equation; The current step size is determined based on the local radius curvature at the current contact point and the preset chord height error:

[0014] In the formula, Ri The local radius of curvature at the current contact point. M The current step size, ε The preset chord height error is 0.01mm. Using the starting point of the optimized polishing path as the origin, and the current step size as the interval, and based on the constraints... x 2 + y 2 ≤400, generate the current discrete point; The current discrete point is used as the next current contact point, and the step size corresponding to the next current contact point is determined. This process is repeated to generate multiple discrete points along the optimized polishing path, thereby obtaining multiple tool point position information for polishing curved workpieces.

[0015] Preferably, the tool point attitude information corresponding to each tool point position information in the workpiece coordinate system is determined based on each tool point position information and a preset precession angle, specifically as follows: The Euler angle corresponding to each tool point position is determined based on the position information of each tool point and the preset precession angle. Based on each Euler angle, determine the tool point posture information corresponding to each tool point position information in the workpiece coordinate system.

[0016] Preferably, the tool point attitude information corresponding to each tool point position information in the workpiece coordinate system is determined based on each Euler angle, specifically as follows: Based on each Euler angle, the corresponding tool point position information is transformed from the tool coordinate system to the workpiece coordinate system to obtain the corresponding rotation matrix; The tool point posture information in the workpiece coordinate system is determined based on each rotation matrix.

[0017] Polishing of curved workpieces is performed based on multiple tool point posture information, specifically as follows: A six-degree-of-freedom robot polishing platform based on a serial robotic arm is used to polish curved workpieces according to multiple tool point posture information; The six-degree-of-freedom robotic polishing platform based on a serial robotic arm includes a clamp, a flat-jaw vise, and a fixed worktable; The fixture includes two symmetrically arranged rectangular clamping members, and the curved workpiece is fixed between the two rectangular clamping members; The flat-nose pliers include two symmetrically arranged rectangular clamps, each rectangular clamping element is respectively disposed on the inner side of the corresponding rectangular clamp, and each rectangular clamp is used to fix the corresponding rectangular clamping element; The fixed workbench is used to place the flat-jaw vise, the clamp, and the curved workpiece.

[0018] Preferably, a six-degree-of-freedom robotic polishing platform based on a serial robotic arm is used to polish the curved workpiece according to multiple tool point posture information, specifically: Any rectangular clamp is designated as the first clamp, and its corresponding rectangular clamping component is designated as the first clamping component. Any vertex outside the first clamp is obtained, and a vertex coordinate system is established with the vertex as the origin. Obtain the geometric parameters of the first clamp, the first clamping member and the fixed worktable, as well as the relative position information of the first vertex with the three axes of the workpiece coordinate system; The translation matrix between the vertex coordinate system and the tool coordinate system is determined based on the geometric parameters and the relative position information; The attitude information is transformed into coordinates based on the translation matrix to obtain the target processing information in the vertex coordinate system; Polish the curved workpiece according to the target processing information.

[0019] Compared with the prior art, the present invention has the following beneficial technical effects: (1) The present invention uses the Newton downhill method to iteratively adjust the initial spacing of adjacent initial path points until the error function converges to the preset allowable error value, and finally determines the optimized polishing path. By dynamically adjusting the spacing of path points, the periodic arrangement mode of the traditional spiral / grating path is broken, and the mid-frequency ripple error caused by periodic motion is suppressed. At the same time, the deterministic optimization mechanism based on error convergence replaces the random layout of pseudo-random paths, avoids the problem of local uneven material removal caused by high randomness, and reduces the excessive dependence on the dynamic performance of ultra-high performance machine tools. (2) The present invention determines the overlap function based on the removal function of two adjacent initial path points, calculates the maximum value of each removal function, and then constructs an error function based on these maximum values, and makes the error function approach 0, ensuring that the optimization objective focuses on making the maximum removal amount of the overlapping area consistent with the maximum removal amount of the non-overlapping area, eliminating the source of difference that may lead to local over-polishing or under-polishing, and ensuring the uniformity of the polished surface. (3) This invention employs Hartley-Judd The method calculates the non-uniformly distributed B-spline node vectors and constructs an interpolation curve equation to generate a continuous optimized polishing path. The discrete optimized path points are smoothly and continuously connected to generate a polishing trajectory that retains the advantages of the low dynamic characteristics of the Archimedes spiral and has the characteristics of non-uniform optimized spacing, effectively avoiding machine tool vibration and polishing marks caused by path discontinuity. (4) By discretizing the optimized polishing path, the present invention achieves adaptive adjustment of the discretization density according to the local geometric complexity (curvature) of the surface. In the region with high curvature (high complexity), a small step size is used to ensure polishing accuracy, and in the region with low curvature (low complexity), a large step size is used to improve processing efficiency, thereby optimizing the calculation and polishing efficiency while ensuring polishing accuracy. (5) The present invention calculates the corresponding Euler angles based on the position information of each discrete tool point and the preset precession angle, and then derives the rotation matrix or quaternion to represent the tool point posture information, thereby realizing the accurate conversion of the theoretically optimized polishing path into motion commands that can be executed by the actual robot polishing platform. (6) The present invention combines the translation matrix to transform the pose information in the tool coordinate system to the vertex coordinate system established based on the vertex of the flat jaw, and compensates for the offset between the theoretical workpiece coordinate system (at the center of the curved surface) and the actual clamping coordinate system (at the vertex of the flat jaw) through coordinate system transformation, thus ensuring the accuracy and real-time of the polishing path in the actual physical space. Attached Figure Description

[0020] Figure 1 This is a flowchart of the polishing path planning method for curved workpieces according to the present invention; Figure 2 This is a schematic diagram of projecting a planar Archimedean spiral onto a curved workpiece in an embodiment of the present invention; Figure 3 This is a schematic diagram of the curved workpiece of the present invention; Figure 4 This is a schematic diagram of the planar Archimedean spiral of the present invention; Figure 5 It is a point i and points i A schematic diagram of overlapping and non-overlapping areas in the initial polishing path of +1; Figure 6 This is a schematic diagram of the tool coordinate system of the present invention; Figure 7 This invention uses ZYX A schematic diagram illustrating the calculation of the rotation matrix using the Euler angles method; Figure 8 This is a front view of a six-degree-of-freedom robotic polishing platform based on a serial robotic arm; Figure 9 This is a side view of a six-degree-of-freedom robotic polishing platform based on a serial robotic arm; Figure 10 This is a schematic diagram of the fabrication structure of the cutting tool (polishing tool) of the present invention; Figure 11 Vertex coordinate system With the tool coordinate system A schematic diagram; In the diagram, 1 is the first clamp; 2 is the first clamping component; 3 is the curved workpiece; 4 is the fixed worktable; 4-1 is the surface of the fixed worktable; 4-2 is the bottom surface of the fixed worktable; 5 is the vertex; and 6 is the cutting tool. Detailed Implementation

[0021] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0022] This invention provides a method for determining the polishing path of a curved workpiece 3, comprising the following steps: S1. Establish a workpiece coordinate system with the center of the curved workpiece 3 as the origin, and determine the initial polishing path based on the Archimedes spiral according to the workpiece coordinate system and the preset spacing parameters. The initial polishing path includes the position information of multiple initial path points and the initial spacing between adjacent initial path points. Preferred, such as Figure 2 As shown, the curved workpiece 3 of the present invention is a parabolic surface, then S1 specifically is: S11, such as Figure 3 As shown, with the center of the curved workpiece 3 as the origin, a workpiece coordinate system is established according to the preset horizontal and vertical axis directions. A planar Archimedean spiral is established directly above the workpiece coordinate system, and its polar coordinate equation is:

[0023] in, r For the rotating polar radius, θ For rotation angle, r 0 is the initial polar radius. b The parameters used to control the spacing of the spiral lines.

[0024] S12. Transform the polar coordinate system equations into parametric equations and project the planar Archimedean spiral path onto the three-dimensional curved workpiece 3 according to the definition of the surface equation to obtain the initial polishing path:

[0025] In the formula, x , y and z These represent the coordinates of any initial path point along different coordinate axes in the workpiece coordinate system.

[0026] Specifically, in step S12, let b =4mm, starting from the origin, generate the planar Archimedean spiral path according to the first two formulas in S12, and then project this planar Archimedean spiral path onto the parabola using the third formula in S12, as follows. Figure 4 As shown, the initial equally spaced Archimedean spiral polishing path is obtained, denoted as the initial polishing path; S13. Based on the Preston model, combined with Hertz contact theory and RBF neural network, determine the radius of the circular contact area between the tool 6 and the curved workpiece 3 at each path point; Furthermore, in this embodiment, as Figure 10 As shown, tool 6 is an elastic polishing tool, made of silicone and 2000-grit diamond sandpaper. This invention is based on the Preston model, combined with Hertz contact theory and RBF neural network. Through finite element simulation, the radius of the circular contact area between the polishing tool and the curved workpiece 3 at different positions during polishing is obtained. Furthermore, an RBF neural network, capable of adapting to nonlinear problems, is used to obtain the "time-varying" circular contact area radius function and the minimum circular contact area radius. In other embodiments, those skilled in the art can also determine the circular contact radius of the tool 6 at different positions of the curved workpiece 3 based on other existing technologies.

[0027] Furthermore, in other embodiments, the present invention also uses the minimum circular contact area radius as a constraint, and employs Newton's descending hill method to control the parameters of the spiral spacing. b The process is iteratively optimized until the end of the initial polishing path reaches the edge of the curved workpiece 3, resulting in the initial equidistant Archimedean spiral, which is denoted as the initial polishing path.

[0028] S2. Select any two adjacent initial path points, denoted as points. i and points i +1, and determine the point. i Removal function and point i +1 is the removal function; In the above-mentioned equally spaced Archimedean spiral polishing path, two adjacent points are randomly selected in sequence. i and i +1. For example... Figure 5 As shown, taking these two points as the origin, calculate the normal vector and tangent vector of the curved workpiece 3 at these two points, and take the principal normal vector of the curved workpiece 3 at these two points as... Z The radial tangent of the surface of the curved workpiece 3 is used as the shaft. X Axis, establish coordinate system O i and O i+1 Assuming at point i The radius of the contact area between the curved workpiece 3 and the polishing tool is... r i At point i The radius of the contact area at +1 is r i+1 ,point i and points i +1 initial path spacing is L i ,from Figure 5 As can be seen, due to the overlap of paths, there are... L i -r i+1 < r i .therefore[- r i , L i -r i+1 ]and[ r i , L i +r i+1 [] represents the non-overlapping region.

[0029] Specifically, the full name of the removal function is the non-overlapping region removal function, and the full name of the overlapping function in the following text is the overlapping region removal function; point i and points i The functions for removing non-overlapping regions with +1 are as follows:

[0030] In the formula, hi For point i The removal function; hi +1 is a point i +1 is the removal function; ri and ri +1 represents points i and points i +1 Radius of the circular contact area between the curved workpiece 3 and the polishing tool; kp The Preston coefficient was determined by establishing a removal function model for tool 6 and using a single-point polishing experiment. The determination process is common knowledge in this field. ω The rotational angular velocity of tool 6; θ The precession angle is a preset value. In this embodiment, the precession angle is the angle between the tool axis and the local normal of the curved workpiece 3 at the center of the contact area. The pressure at the center of the circular contact area is obtained through stress distribution analysis in the contact area and the following formula:

[0031] in, F n This refers to the normal force applied during the polishing process; δ The downward pressure of tool 6 can be obtained from the following formula:

[0032] in, R t Let be the radius of the cutting tool.

[0033] S3, based on point i Removal function and point i The +1 removal function determines the error function, and based on the error function and the point... i and points i The initial spacing of +1 determines the adjustment amount in Newton's descent method; S3 specifically refers to: S31, According to the point i Removal function and point i The removal function with +1 determines the overlapping function; The function for removing the overlapping region between two points is:

[0034] S32, Determine the point i The maximum value and point of the removal function i The maximum value of the function with +1 is denoted as the first maximum value and the second maximum value, respectively. In one embodiment, the first maximum value is determined using the Preston model. H i Second maximum value H i+1 and the maximum value of the overlapping function H i，i+1 .

[0035] S33. Determine the error function based on any two of the first maximum value, the second maximum value, and the overlap function.

[0036] Specifically, the overlapping function h i，i+1 Maximum value and removal function h i The mean square error of the maximum value is used as the error function;

[0037] In the formula, E This is the error function.

[0038] S34. Based on the error function and points i and points i The initial spacing of +1 determines the adjustment amount in Newton's descent method. Specifically, the adjustment amount and learning rate η in Newton's descent method should satisfy:

[0039] S4. Iteratively adjust the midpoint of the initial polishing path using Newton's downhill method. i and points i An initial spacing of +1 is used to obtain the point when the error function converges to a preset allowable error value. i and points i +1 optimized polishing spacing;

[0040] In the formula, m This represents the number of iterations. and The first m Second and third m +1 The polishing spacing obtained from iterative calculation; This is the adjustment amount in Newton's descent method; This is an empirical parameter that can be adjusted according to specific problems.

[0041] Iterative calculations based on the above formula continue until... E It is less than the preset allowable error value. Specifically, the allowable error value is determined based on the difference between the first maximum value and the third maximum value.

[0042] Preferably, in one embodiment, the application will satisfy the following conditions when the corresponding E * The value of is used as the allowable error value:

[0043] at this time E * The following conditions must be met:

[0044] In the formula, E * This is the allowable error value.

[0045] The following is a detailed explanation of the principle of this invention: During the polishing process, the contact areas formed by the polishing tool at two adjacent polishing points are called non-overlapping areas. When the path spacing is small, these non-overlapping areas will partially overlap due to the continuous movement of the tool, forming overlapping areas. The amount of material removed from the non-overlapping areas is mainly determined by the depth of a single polishing pass, while the amount removed from the overlapping areas is the cumulative result of multiple polishing effects. If the maximum removal amounts of the non-overlapping and overlapping areas are inconsistent, the surface material will exhibit an uneven phenomenon of "some places being polished deeply, and some places being polished shallowly." This invention adjusts the path spacing to make the maximum removal amounts of the overlapping and non-overlapping areas equal, which is equivalent to setting a uniform "removal benchmark" for the entire surface. That is, whether it is the non-overlapping area of ​​a single polishing pass or the overlapping area of ​​multiple passes, the depth of material removal is controlled at the same level. This uniformity avoids local over-polishing or under-polishing, fundamentally eliminating the mid-frequency error caused by differences in removal amounts, and ultimately achieving precise control of surface quality.

[0046] S5. Similarly, determine the polishing spacing of the remaining adjacent initial path points, and use the polishing spacing of multiple adjacent points to adjust the position information of each initial path point to obtain multiple optimized path points in the workpiece coordinate system. Specifically, in the context of i and i After optimizing the initial polishing spacing of +1, select the next pair of adjacent initial path points and adjust their initial polishing spacing. When all adjacent path points on the current radial line have been optimized, save the path points on this radial line, select the next radial line, and repeat the above process until the initial polishing path is optimized.

[0047] This invention sets the optimization objective to make E Approaching 0 optimizes the uniformity of material removal between overlapping and non-overlapping areas of the polishing path, thereby suppressing mid-frequency errors.

[0048] S6. Determine the polishing path of the curved workpiece 3 based on multiple optimized path points.

[0049] S61. Based on the Euclidean distance between adjacent optimized path points, the following is adopted: Hartley-Judd The method calculates the nodal vectors of the B-spline interpolation curve to obtain the non-uniformly distributed nodal values; S62. Construct interpolation curve equations based on node values ​​to generate continuous optimized polishing paths, and use the optimized polishing paths as the polishing paths for surface workpiece 3.

[0050] Specifically, node values u The calculation formula is as follows:

[0051] In the formula, k for Hartley-Judd The order of the B-spline curve in the method is, in this case, a cubic B-spline curve. k The value is 4; n To optimize the total number of path points. i To optimize the index variables of path points, the range is... k +1 to n ; u i For the first i The node values ​​corresponding to each optimized path point u i It is a normalized parameter calculated by accumulating the chord lengths (or Euclidean distances) between adjacent path points.

[0052] This invention uses a non-uniform cubic B-spline interpolation curve to connect the remaining path points into a polishing path similar to an Archimedean spiral, and uses... Hartley-Judd The method for node vectors u Calculations are performed to obtain the node values ​​and generate an optimized polishing path map.

[0053] This invention also discloses a polishing method for a curved workpiece 3, applied to the polishing path determined by the polishing path determination method for the curved workpiece 3. The polishing method includes the following steps: like Figure 6 As shown, a tool coordinate system coinciding with the workpiece coordinate system is established with the starting point of the initial polishing path as the origin. Specifically, the three axes of the tool coordinate system and the workpiece coordinate system are in the same direction, but their origins are different.

[0054] The optimized polishing path is discretized to obtain multiple tool point position information for polishing the curved workpiece 3 in the tool coordinate system; The optimized polishing path is discretized using the equal chord height error method to obtain multiple tool point position information for polishing the curved workpiece 3 in the tool coordinate system, specifically: The local radius curvature at the current contact point between the workpiece 3 and the tool 6 is determined based on the surface equation:

[0055]

[0056] In the formula, R i The local radius of curvature at the current contact point. x and y These are the x and y coordinates of the current contact point in the tool coordinate system, with the first current contact point being the starting point of the optimized polishing path. z The definition of the surface equation for a curved workpiece; The current step size is determined based on the local radius curvature at the current contact point and the preset chord height error:

[0057] In the formula, Ri The local radius of curvature at the current contact point. M The current step size, ε The preset chord height error is 0.01mm. Using the starting point of the optimized polishing path as the origin, and the current step size as the interval, and based on the constraints... x 2 + y 2 ≤400, generate the current discrete point; The current discrete point is used as the next current contact point, and the step size corresponding to the next current contact point is determined. This process is repeated to generate multiple discrete points along the optimized polishing path, thereby obtaining multiple tool point position information for polishing the curved workpiece 3.

[0058] Based on the tool point position information and the preset precession angle, determine the tool point attitude information corresponding to each tool point position information in the workpiece coordinate system; Based on the tool point position information and the preset precession angle, the tool point attitude information corresponding to each tool point position information in the workpiece coordinate system is determined, specifically as follows: In this invention, the tool point position information refers to the coordinates of the tool center point in the tool coordinate system. X, Y, Z Tool point posture information refers to the direction of the tool axis.

[0059] The Euler angle corresponding to each tool point position is determined based on the position information of each tool point and the preset precession angle. like Figure 7 As shown, X T Axis projection onto surface X W O W Y W The projection line passes through the point O W ,point A For a point on the projection line, we can obtain Z T Angle of rotation of the shaft α for:

[0060] In the formula, xi , yi and zi The results obtained in step eight are as follows: i The tool point position information corresponding to the point.

[0061] Z T Translate axis to point O W get Z T 'axis, Y T Axis rotation angle β for:

[0062] In the formula, xi , yi and zi The results obtained in step eight are as follows: i The corresponding tool point position information; θ The precession angle is a preset value. In this embodiment, the precession angle is the angle between the axis of the polishing tool and the local normal of the curved workpiece 3 at the center of the contact area.

[0063] In this embodiment, θ The default value is 20°.

[0064] tool coordinate system O T No relative X T The axis rotates, therefore γ The angle is 0.

[0065] Based on each Euler angle, determine the tool point posture information corresponding to the tool point position information in the workpiece coordinate system.

[0066] In other embodiments, each Euler angle can be directly used to represent the corresponding tool point posture information.

[0067] Preferably, in this embodiment, the corresponding tool point position information is also transformed from the tool coordinate system to the workpiece coordinate system based on each Euler angle to obtain the corresponding rotation matrix: like Figure 7 As shown, maintain the tool coordinate system O T of X T The shaft and the curved workpiece 3 are tangentially coplanar at the center of the circular contact area, and use... ZYX Euler angle method for calculating rotation matrix R， Assuming tool coordinate system O T With workpiece coordinate system O W Overlap, and sequentially adjust the tool coordinate system O T Around Z T Rotation α Corner, around Y T Rotation β Corner, around X T Rotation γ The resulting rotation matrix is:

[0068] The result of the rotation matrix is ​​as follows:

[0069] In the formula: c cos, an abbreviation for cosine angle; s sin is an abbreviation for sine angle. α horn, β Angle and γ All angles are Euler angles, which can be calculated using the formula mentioned above.

[0070] The tool point posture information in the workpiece coordinate system is determined based on each rotation matrix.

[0071] In other words, this invention calculates the corresponding Euler angles based on the position information of each tool point, and then calculates the corresponding rotation matrix based on each Euler angle. The rotation matrix can be directly used to represent the tool point attitude information. For each optimized path point, there are corresponding tool point position information, Euler angles, and rotation matrix.

[0072] In other embodiments, tool point orientation information can also be represented using quaternions; the derivation process of quaternions is as follows:

[0073]

[0074]

[0075]

[0076]

[0077] In the formula, R It is a rotation matrix; , , , It is a quaternion.

[0078] The above derivation process is common knowledge in this field and will not be elaborated further; from the above rotation process, it can be seen that: in order to calculate the quaternion, the position information of each tool point must first be solved from the tool coordinate system. O T Transform to workpiece coordinate system O W By using the rotation matrix and the conversion relationship between quaternions and rotation matrices, the quaternion corresponding to each tool point position information can be obtained; the quaternion can be used to represent the tool point attitude information.

[0079] Polishing of curved workpiece 3 is performed based on multiple tool point posture information.

[0080] The method in this embodiment is applicable to a six-degree-of-freedom robotic polishing platform based on a serial robotic arm, including an ABB IRB4600-60 / 2.05 robotic arm, a fixed worktable 4, a flat-jaw vise, a special curved workpiece 3 fixture, and an asynchronous three-phase AC spindle.

[0081] like Figure 8 As shown, the fixture includes two symmetrically arranged rectangular clamping members, and the curved workpiece 3 is fixed between the two rectangular clamping members; The flat-nose pliers include two symmetrically arranged rectangular clamps, each rectangular clamping element is respectively set on the inner side of the corresponding rectangular clamp, and each rectangular clamp is used to fix the corresponding rectangular clamping element; The fixed workbench 4 is used to place flat-jaw vises, clamps, and curved workpieces 3.

[0082] Since the center of the aspherical surface workpiece 3 has no sharp point or straight edge, it is impossible to establish the workpiece coordinate system at the center of the aspherical surface workpiece 3 in the actual robot system. Instead, the workpiece coordinate system can only be established at the vertex 5 of any clamp of the flat jaw vise. Therefore, the pose of the tool position point of the polishing path needs to be adjusted according to the actual size parameters of the polishing platform.

[0083] Then we have: A six-degree-of-freedom robotic polishing platform based on a serial robotic arm is used to polish the curved workpiece 3 according to multiple tool point posture information. Specifically: Let any rectangular clamp be denoted as the first clamp 1, and its corresponding rectangular clamping component be denoted as the first clamping component 2. Obtain any vertex 5 outside the first clamp 1, and establish a vertex coordinate system with vertex 5 as the origin. like Figure 11 The diagram shows the established vertex coordinate system and tool coordinate system; the vertex coordinate system is then determined. With the tool coordinate system Translation matrix:

[0084] In the formula: It is a translation matrix. It is along x The distance between vertex 5 and the origin of the workpiece coordinate system in the direction of axis extension. x 1 is the distance from vertex 5 to the first contact surface, which is the contact surface between the first clamp 1 and the first clamping member 2. x 2 is the width of the first clamping member 2. x 3 represents the distance between the two clamping parts. It is along y The distance between vertex 5 and the origin of the workpiece coordinate system in the direction of axis extension. y 1 is the length of the first clamping member 2 along the extending direction. It is along z The distance between vertex 5 and the origin of the workpiece coordinate system in the direction of axis extension. z 1 is the distance from the surface 4-1 of the fixed worktable to the bottom of the curved workpiece 3. z 2 is the height of the curved workpiece 3. z 3 is the distance from the bottom surface 4-2 of the fixed workbench to the surface 4-1 of the fixed workbench. z 4 is the distance from the bottom surface 4-2 of the fixed workbench to the upper surface of the first rectangular clamp.

[0085] like Figure 11 As shown, correspondingly, when implementing the present invention, one can obtain and The specific location.

[0086] The attitude information is transformed into coordinates using the translation matrix to obtain the target processing information in the vertex coordinate system; Specifically, Euler angles, rotation matrices, or quaternions can be multiplied by translation matrices to obtain the translated attitude information, which is recorded as the target processing information.

[0087] Polish the curved workpiece 3 according to the target machining information.

[0088] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

Claims

1. A method for determining the polishing path of a curved workpiece, characterized in that, Includes the following steps: S1. Establish a workpiece coordinate system with the center of the curved workpiece as the origin, and determine an initial polishing path based on the Archimedes spiral according to the workpiece coordinate system and the preset spacing parameters. The initial polishing path includes the position information of multiple initial path points and the initial spacing between adjacent initial path points. S2. Select any two adjacent initial path points and determine their respective removal functions, denoted as the first removal function and the second removal function, respectively. S3. Determine the error function based on the first removal function and the second removal function, and determine the adjustment amount in Newton's downhill method based on the error function and the initial distance between two adjacent initial path points; S4. Iteratively adjust the initial spacing between two adjacent initial path points in the initial polishing path according to the adjustment amount until the error function converges to the preset allowable error value, and obtain the optimized polishing spacing between the two adjacent initial path points. S5. Following the above method, determine the optimized polishing spacing of the remaining adjacent initial path points, and use multiple optimized polishing spacings to adjust the position information of each initial path point to obtain multiple optimized path points in the workpiece coordinate system. S6. Determine the polishing path of the curved workpiece based on the multiple optimized path points.

2. The method for determining the polishing path of a curved workpiece according to claim 1, characterized in that, S3 specifically refers to: S31. Determine the overlap function based on the first removal function and the second removal function; S32. Determine the maximum value of the first removal function and the maximum value of the second removal function, and denot them as the first maximum value and the second maximum value, respectively; S33. Determine the error function based on any two of the first maximum value, the second maximum value, and the overlap function.

3. The method for determining the polishing path of a curved workpiece according to claim 2, characterized in that, S33 specifically refers to: S331. Determine the maximum value of the overlap function, denoted as the third maximum value; S332. Determine the error function based on the root mean square deviation of any two of the first maximum value, the second maximum value, and the third maximum value.

4. The method for determining the polishing path of a curved workpiece according to claim 1, characterized in that, S6 specifically refers to: S61. Based on the Euclidean distance between adjacent optimized path points, the following method is adopted: Hartley-Judd The method calculates the nodal vectors of the B-spline interpolation curve to obtain the non-uniformly distributed nodal values; S62. Construct an interpolation curve equation based on the node values ​​to generate a continuous optimized polishing path, and use the optimized polishing path as the polishing path of the curved workpiece.

5. A polishing method for curved workpieces, characterized in that, The polishing path determined by the polishing path determination method for curved workpieces according to claim 4, wherein the polishing method includes the following steps: A tool coordinate system coinciding with the workpiece coordinate system is established with the starting point of the initial polishing path as the origin. The optimized polishing path is discretized to obtain multiple tool point position information for polishing curved workpieces in the tool coordinate system; Based on the tool point position information and the preset precession angle, determine the tool point attitude information corresponding to each tool point position information in the workpiece coordinate system; Polishing of curved workpieces is performed based on multiple tool point posture information.

6. The polishing method for curved workpieces according to claim 5, characterized in that, The optimized polishing path is discretized to obtain multiple tool point position information for polishing curved workpieces in the tool coordinate system, specifically: Determine the local radius curvature at the current contact point between the workpiece and the tool based on the surface equation: In the formula, R i The local radius of curvature at the current contact point; x and y These are the x and y coordinates of the current contact point in the tool coordinate system, with the first current contact point being the starting point of the optimized polishing path. z This is the definition of the surface equation; The current step size is determined based on the local radius curvature at the current contact point and the preset chord height error: In the formula, Ri The local radius of curvature at the current contact point. M The current step size, ε The preset chord height error is 0.01mm. Using the starting point of the optimized polishing path as the origin, and the current step size as the interval, and based on the constraints... x 2 + y 2 ≤400, generate the current discrete point; The current discrete point is used as the next current contact point, and the step size corresponding to the next current contact point is determined. This process is repeated to generate multiple discrete points along the optimized polishing path, thereby obtaining multiple tool point position information for polishing curved workpieces.

7. The polishing method for curved workpieces according to claim 6, characterized in that, Based on the tool point position information and the preset precession angle, the tool point attitude information corresponding to each tool point position information in the workpiece coordinate system is determined, specifically as follows: The Euler angle corresponding to each tool point position is determined based on the position information of each tool point and the preset precession angle. Based on each Euler angle, determine the tool point posture information corresponding to each tool point position information in the workpiece coordinate system.

8. The polishing method for curved workpieces according to claim 7, characterized in that, Based on each Euler angle, the tool point orientation information corresponding to each tool point position information in the workpiece coordinate system is determined, specifically as follows: Based on each Euler angle, the corresponding tool point position information is transformed from the tool coordinate system to the workpiece coordinate system to obtain the corresponding rotation matrix; The tool point posture information in the workpiece coordinate system is determined based on each rotation matrix.

9. The polishing method for curved workpieces according to claim 8, characterized in that, Polishing of curved workpieces is performed based on multiple tool point posture information, specifically as follows: A six-degree-of-freedom robot polishing platform based on a serial robotic arm is used to polish curved workpieces according to multiple tool point posture information; The six-degree-of-freedom robotic polishing platform based on a serial robotic arm includes a clamp, a flat-jaw vise, and a fixed worktable; The fixture includes two symmetrically arranged rectangular clamping members, and the curved workpiece is fixed between the two rectangular clamping members; The flat-nose pliers include two symmetrically arranged rectangular clamps, each rectangular clamping element is respectively disposed on the inner side of the corresponding rectangular clamp, and each rectangular clamp is used to fix the corresponding rectangular clamping element; The fixed workbench is used to place the flat-jaw vise, the clamp, and the curved workpiece.

10. The polishing method for curved workpieces according to claim 9, characterized in that, A six-DOF robotic polishing platform based on a serial robotic arm is used to polish curved workpieces according to multiple tool point posture information. Specifically: Any rectangular clamp is designated as the first clamp, and its corresponding rectangular clamping component is designated as the first clamping component. Any vertex outside the first clamp is obtained, and a vertex coordinate system is established with the vertex as the origin. Obtain the geometric parameters of the first clamp, the first clamping member, and the fixed worktable, as well as the relative position information of any vertex with respect to the three axes of the workpiece coordinate system; The translation matrix between the vertex coordinate system and the tool coordinate system is determined based on the geometric parameters and the relative position information; The attitude information is transformed into coordinates based on the translation matrix to obtain the target processing information in the vertex coordinate system; Polish the curved workpiece according to the target processing information.

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