A free-form surface polishing system and curvature adaptive polishing method based on a robotic arm

Abstract

This invention discloses a freeform surface polishing system and a curvature-adaptive polishing method based on a robotic arm. The freeform surface polishing system includes a six-degree-of-freedom robotic arm, an end-effector polishing head, a six-dimensional force sensor, a worktable, and a fixture. The end-effector polishing head is driven by a motor and fixed to the end of the six-degree-of-freedom robotic arm. The six-dimensional force sensor is installed at the bottom of the end-effector polishing head. The fixture is fixed to the worktable by bonding or welding, and the worktable is located within the nominal working range of the robotic arm. The curvature-adaptive polishing method includes a high-precision surface polishing method combining macroscopic trajectory planning and microscopic material removal for freeform surfaces, and a posture control method for the robotic arm polishing freeform surfaces. This invention overcomes the problems of high cost, large size, and numerous limitations associated with traditional machine tool processing of freeform surface components, making the polishing of freeform surface components more efficient, improving the production accuracy of parts, and ensuring product quality.

Description

Technical Field

[0001] This invention belongs to the field of precision and ultra-precision machining of robotic arms, specifically relating to a free-form surface polishing system and a curvature adaptive polishing method based on a robotic arm. Background Technology

[0002] With the development of information technology and automation technology, the processing of curved surfaces is becoming increasingly stable and efficient, and more and more new technologies and methods are being applied to curved surface processing. Among them, free-form surface polishing with robotic arms is an important way to obtain high-precision optical components, especially in the fields of electronics, lasers, and aerospace.

[0003] The polishing and removal system for optical freeform surfaces is a complex multi-input multi-output system. Current research on material removal models and freeform surface polishing compensation in this field suffers from drawbacks such as low accuracy, cumbersome operation, and high cost. Conventional small-segment interpolation techniques can achieve geometrical continuity of the machining trajectory and are widely used in surface machining; however, there is no good method to accurately obtain the optimal solution for freeform surfaces. Furthermore, the application of robotic arms for freeform surface polishing in the industry faces the challenge of poor accuracy. Summary of the Invention

[0004] To address the issue of poor precision in machining high-precision freeform surface components using robotic arms in existing industries, this invention provides a robotic arm-based freeform surface polishing system and a curvature-adaptive polishing method. Based on a novel material polishing and removal mechanism, this invention proposes a corresponding freeform surface machining path planning approach; develops a novel machining compensation method based on a six-dimensional force sensor; and designs corresponding hardware equipment, including tooling and adjustment. This overcomes the challenges of high cost, large size, and numerous limitations associated with traditional machine tool machining of freeform surface components, making the polishing of freeform surface components more efficient, improving the production precision of parts, and ensuring product quality.

[0005] To achieve the above objectives, the present invention employs the following technical solution:

[0006] A free-form surface polishing system based on a robotic arm includes a six-degree-of-freedom robotic arm, an end-effector polishing tool head, a six-dimensional force sensor, a worktable, and a tooling fixture;

[0007] The end-effector polishing head is driven by a motor and fixed to the end of the six-degree-of-freedom robotic arm; a six-dimensional force sensor is installed at the bottom of the end-effector polishing head; the tooling fixture is fixed to the worktable by bonding or welding, and the worktable is located within the nominal working range of the robotic arm.

[0008] A further improvement of this invention is that the polishing path motion of the free-form surface is completed by programming a six-degree-of-freedom robotic arm. When the worktable is located in the middle of the motion range, the whole system is in the optimal working state, with the best vibration state and motion accuracy. At the same time, based on the function of the six-dimensional force sensor to provide real-time feedback of three-dimensional force and three-dimensional torque, as well as the mechanical properties of the polishing head, a repeatability positioning accuracy of 0.02mm is achieved.

[0009] A further improvement of the present invention is that the six-dimensional force sensor is positioned at the end of the polishing tool head to collect the normal polishing force of the polishing head. The data collected by the six-dimensional force sensor is fed back to the compensating motion of the robotic arm, thereby forming a closed-loop control to compensate for the error of the polishing force.

[0010] A further improvement of this invention is that the polishing force at the end is collected by a six-dimensional force sensor, and the force-displacement compression performance of the polishing head is fed back in real time to the displacement control of the robotic arm end perpendicular to the processing surface, thereby compensating for the error of the robotic arm pressure control, and thus compensating for the influence of polishing pressure on the indentation of polishing particles and the actual contact area, that is, compensating for the indentation coefficients k2 and k3 mentioned below. When the pressure is low, it is assumed that the indentation coefficients change linearly with the polishing pressure.

[0011] A further improvement of the present invention is that the tooling fixture includes a supporting base plate and an upper grooved fixture; the supporting base plate has air suction holes, and the workpiece is fixed by vacuum suction and bonding; one side of the upper grooved fixture is adjustable and the other side is fixed.

[0012] A further improvement of the present invention is that a ball screw is provided inside the support base plate, and a movable baffle is threaded onto the ball screw. The linear motion of the movable baffle can be completed by the rotational motion of the ball screw.

[0013] A curvature-adaptive polishing method based on a robotic arm, the method being based on the aforementioned freeform surface polishing system based on a robotic arm, includes the following steps:

[0014] 1) A high-precision surface polishing method combining macroscopic trajectory planning and microscopic material removal for freeform surfaces: including a multi-coefficient polishing removal mechanism and trajectory planning for robotic arm adaptive freeform surface polishing based on NURBS surface curvature parameters; based on the Preston equation and the robotic arm's geometric motion model, a variable coefficient polishing removal mechanism formula is proposed according to the abrasive embedding mechanism and the rough material contact mechanism; based on the characteristics of NURBS surfaces, an efficient and high-precision variable coefficient polishing trajectory planning method for robotic arm polishing of freeform surfaces is proposed, and an adaptive machining trajectory spacing is planned according to the curvature change of the freeform surface;

[0015] 2) Pose control method for a robotic arm to polish a free-form surface: The motion control for polishing the free-form surface is carried out by fitting the surface with point clouds. The specific polishing motion is realized based on the kinematics of a six-degree-of-freedom robotic arm, including point cloud position motion and optimal pose control. For pose control, two principles are proposed: adopting a pose with the maximum possible polishing efficiency and no interference with the surface to improve accuracy and efficiency.

[0016] A further improvement of the present invention is that in step 1), the specific implementation method is as follows:

[0017] 101) Propose the relative motion of the robotic arm with respect to the workpiece during polishing, and calculate the relative motion speed of the abrasive grain with respect to the workpiece according to the relative motion speed of the robotic arm and the workpiece; Since there is also relative motion between the abrasive grain and the robotic arm, a speed coefficient k1 is introduced to characterize the relative motion speed of the polishing abrasive grain and the robotic arm, that is, v2 = k1v1, where v1 is the relative motion speed of the robotic arm and the workpiece, v2 is the relative motion speed of the abrasive grain and the workpiece, and 0 < k1 < 1, that is, the relative motion speed of the polishing abrasive grain and the workpiece is less than the relative motion of the robotic arm and the workpiece;

[0018] 102) Based on the micro-mechanism of abrasive grain embedding and the contact mechanism of macroscopic rough materials, propose the Preston material removal equation;

[0019] Among them, the depth of the abrasive grain embedded in the workpiece is determined by the pressure of the robotic arm and the inherent properties of the material, and its size is less than the diameter D of the abrasive grain. Therefore, an embedding coefficient k2 is introduced to characterize the influence of the pressure of the robotic arm and the material on the embedding depth, that is, the abrasive grain embedding depth is k2D, and 0 < k2 < 1, and its size is related to the pressure of the robotic arm and the material properties, and is obtained through parameter experiments; When rough materials come into contact with each other, the actual contact area will be less than the total area. Therefore, a contact coefficient k3 is introduced to characterize the contact error of the rough surface, that is, the actual contact area A = k3A0. Similarly, 0 < k3 < 1; In addition, when the pressure is small, it is considered that k2 and k3 change linearly with the change of pressure; A six-axis force sensor is used to compensate the pressure of the robotic arm, and the coefficients k2 and k3 change linearly according to the compensated pressure;

[0020] Among them, the removal rate of a single micro-abrasive grain is:

[0021]

[0022] In the formula, ΔS is the particle embedding area, with the unit of mm 2 , δ w is the particle embedding depth, with the unit of mm; Since the polishing head is relatively flexible, the depth δ w of the particle embedding element is much smaller than D, which is simplified to:

[0023] Therefore, the macroscopic polishing material removal rate ρ is:

[0024]

[0025] In the formula, N A This represents the total number of abrasive grains, which is related to the concentration of the polishing slurry.

[0026] 103) Calculate the row spacing of the macroscopic polishing trajectory, and obtain the curvature-adaptive row spacing by controlling the residual height; fit the freeform surface to be processed into a NURBS surface, and the rational fraction of the surface is:

[0027]

[0028] In the formula, u and v are parameter variables on the surface; P ij These are control points on the surface; i and j represent the number of meshes; ω ij These are the weight factors associated with the control vertex; p and q are the number of weight factors; N i，p (u), N j，q (v) are defined as nonrational B-spline basis functions on the node vectors U and V, respectively;

[0029] Furthermore, given a freeform NURBS surface, the curvature data at any point can be obtained;

[0030] The contact area between the polishing tool head and the component is considered as an ellipse. The polishing process of the polishing tool head polishing the variable curvature component is modeled. A coordinate system is established for the contour of the polishing tool head. The row spacing between the two polishing paths is K. Then the residual height in the middle is h, and h represents the surface quality after processing.

[0031] The formula for an ellipse is:

[0032]

[0033] In the formula, a and b are ellipse parameters, that is, the geometric contour parameters of the polishing tool. Different polishing tool heads will have different contour parameter formulas.

[0034] The implicit formula for calculating line spacing is proposed as follows:

[0035]

[0036] For a plane, K = 2x, y = h; for a convex surface, K1 = k4 * K, and k4 < 1; for a concave surface, K1 = k4 * K, and k4 > 1.

[0037] Based on the line spacing with different curvatures, a novel adaptive line spacing path planning method combining curvature is obtained;

[0038] The path is planned based on the macroscopic optimal row spacing, and the process parameters are planned based on the microscopic removal mechanism. By controlling these two aspects, a high-precision freeform surface can be obtained.

[0039] A further improvement of the present invention is that, in step 2), the specific implementation method is as follows:

[0040] The trajectory planning of freeform surfaces adopts a single-point motion mode controlled by point cloud or a circular fitting motion mode, decomposing the motion of complex surfaces into single-point motions, with the position of each point determined by contact pressure and NURBS curvature. A motion control method for freeform surface polishing is proposed by using point cloud fitting of the surface, and the specific polishing motion is realized based on the kinematics of a six-degree-of-freedom robotic arm, including point cloud position motion and optimal attitude control, to improve accuracy and efficiency.

[0041] A further improvement of this invention is that the origin of the end-effector coordinate system of the robotic arm is set as the center of the spherical polishing head, and the point cloud position is calculated by the surface normal and the radius of the polishing head to compensate for overcompression, thereby obtaining the coordinates of the center of the spherical polishing head of the robotic arm when polishing a certain point on the surface:

[0042]

[0043] In the formula, x0 and y0 are points on the freeform surface, R is the radius of the spherical polishing head that has been compensated for the compression of the polishing head, and y′(x0) is the derivative of the explicit expression, corresponding to the slope parameter of the freeform surface;

[0044] Two principles are proposed for attitude control: a set tilt angle to ensure the self-rotation linear velocity at the contact position to improve polishing efficiency and to avoid interference with the curved surface. Therefore, based on the curvature variation law of the spherical polishing head, 45° is selected as the optimal tilt angle.

[0045] Among them, because Euler angles have a gimbal lock, quaternions are chosen to specifically control the posture of the robotic arm.

[0046] Compared with the prior art, the present invention has at least the following beneficial technical effects:

[0047] This invention provides a high-precision freeform surface polishing system based on a robotic arm, incorporating a novel end-material polishing removal mechanism. Based on this mechanism, a corresponding freeform surface processing technology is proposed, thus providing a scientific theoretical foundation for achieving high-precision polishing. A polishing force compensation processing method based on a six-dimensional force sensor is designed, along with corresponding hardware equipment including tooling and adjustment. Errors are comprehensively considered from tooling to processing equipment, thus providing a technological basis for achieving high-precision polishing. This system improves the accuracy and efficiency of high-precision freeform surface polishing, proposes a high-precision polishing removal mechanism model, and designs a fixture device for specific freeform surface workpieces. Attached Figure Description

[0048] Figure 1This is a free-form surface polishing system model in an embodiment of the present invention;

[0049] Figure 2 This is a fixture model in an embodiment of the present invention;

[0050] Figure 3 This is the macroscopic trajectory planning and microscopic particle removal model in the embodiments of the present invention;

[0051] Figure 4 This is a schematic diagram illustrating the optimized posture of the polishing head in an embodiment of the present invention;

[0052] Figure 5 This is a planetary trajectory model in an embodiment of the present invention;

[0053] Figure 6 This is a block diagram of the point cloud generation and search algorithm in an embodiment of the present invention.

[0054] Explanation of reference numerals in the attached figures:

[0055] 1-End polishing tool head, 2-Six-dimensional force sensor, 3-Six-degree-of-freedom robotic arm, 4-Tooling fixture, 5-Worktable, 6-Support base plate, 7-Fixing bolt, 8-Modible baffle, 9-Ball screw, 10-Air suction hole. Detailed Implementation

[0056] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided to enable a more thorough understanding of the present disclosure and to fully convey the scope of the disclosure to those skilled in the art. It should be noted that, unless otherwise specified, the embodiments and features described herein can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0057] The present invention provides a freeform surface polishing system based on a robotic arm, comprising a six-degree-of-freedom robotic arm 3, an end-effector polishing head 1, a six-dimensional force sensor 2, a worktable 5, and a fixture 4; wherein the end-effector polishing head 1 is driven by a motor and fixed to the end of the six-degree-of-freedom robotic arm 3; the six-dimensional force sensor 2 is installed at the bottom of the end-effector polishing head 1; the fixture 4 is fixed to the worktable 5 by bonding or welding, and the worktable 5 is located within the nominal working range of the robotic arm.

[0058] Specifically, the polishing path motion for the free-form surface is programmed using a six-degree-of-freedom robotic arm. It is proposed that the overall system operates optimally when the worktable is positioned at the midpoint of the motion range, exhibiting good vibration control and motion accuracy. Based on the real-time feedback of three-dimensional force and torque from the six-dimensional force sensor and the mechanical properties of the polishing head, a repeatability accuracy of 0.02mm is achieved.

[0059] The six-dimensional force sensor is positioned at the end of the polishing tool head to collect the normal polishing force of the polishing head. The data collected by the sensor is fed back to the compensating movement of the robotic arm, thus forming a closed-loop control to compensate for the error in polishing force. Specifically, the polishing force at the end of the tool head is collected by the six-dimensional force sensor, and the force-displacement compression performance of the polishing head is fed back in real time to the displacement control of the robotic arm end perpendicular to the processing surface. This compensates for the error in the pressure control of the robotic arm, thereby compensating for the influence of polishing pressure on the indentation of polishing particles and the actual contact area, i.e., compensating for the indentation coefficients k2 and k3 mentioned below. When the pressure is low, the indentation coefficients are approximately considered to change linearly with the polishing pressure.

[0060] Based on the characteristics of freeform surface clamping, this invention designs an adjustable clamping device, as shown in the attached figure. Figure 2 As shown, the system specifically includes a support base plate 6 and an upper slotted fixture. The support base plate 6 is connected to the air suction base by four fixing bolts 7. An internal ball screw 9 is added to the support base plate 6, and the linear movement of the movable baffle 8, which is threadedly connected to the ball screw 9, is achieved through rotational motion. The support base plate 6 has air suction holes 10, allowing for workpiece fixation using vacuum suction and bonding methods. One side of the upper slotted fixture is adjustable, while the other side is fixed.

[0061] This invention provides a curvature adaptive polishing method based on a robotic arm. This method, based on the aforementioned freeform surface polishing system based on a robotic arm, includes the following steps:

[0062] 1) A high-precision surface polishing method combining macroscopic trajectory planning and microscopic material removal for freeform surfaces: This includes a novel multi-coefficient polishing removal mechanism and a trajectory planning method for robotic arm adaptive freeform surface polishing based on NURBS surface curvature parameters. Based on the Preston equation and the robotic arm's geometric motion model, a novel variable-coefficient polishing removal mechanism formula is proposed according to the abrasive embedding mechanism and the rough material contact mechanism. Based on the characteristics of NURBS surfaces, an efficient and high-precision variable-coefficient polishing trajectory planning method for robotic arm polishing of freeform surfaces is proposed, and the adaptive machining trajectory spacing is planned according to the curvature change of the freeform surface.

[0063] 2) Pose control method for robotic arm polishing of free-form surfaces: A motion control method for free-form surface polishing is proposed by fitting the surface with point clouds. Based on the kinematics of a six-axis robotic arm, specific polishing motions are realized, including point cloud position motion and optimal pose control. For pose control, two principles are proposed to improve accuracy and efficiency: maximizing the polishing efficiency and ensuring non-interference with the surface.

[0064] A high-precision surface polishing method that combines macroscopic trajectory planning and microscopic material removal for free-form surfaces has the following key points:

[0065] 101) It is proposed that the robotic arm moves relative to the workpiece during polishing, and the relative motion speed of the abrasive grains to the workpiece is calculated based on the relative motion speed of the robotic arm to the workpiece. Since there is also relative motion between the abrasive grains and the robotic arm, a velocity coefficient k1 is introduced to characterize the relative motion speed of the polishing abrasive grains to the robotic arm, that is, v2 = k1v1, where v1 is the relative motion speed of the robotic arm to the workpiece, and v2 is the relative motion speed of the abrasive grains to the workpiece. And 0 < k1 < 1, that is, the relative motion speed of the polishing abrasive grains to the workpiece is slightly less than the relative motion of the robotic arm to the workpiece. The specific value is related to the material and can be obtained through parameter experiments.

[0066] 102) Based on the microscopic mechanism of abrasive grain embedding and the contact mechanism of macroscopic rough materials, a new Preston material removal equation is proposed.

[0067] Among them, the depth of the abrasive grain embedded in the workpiece is determined by the pressure of the robotic arm and the inherent properties of the material, and its size is less than the diameter D of the abrasive grain. Therefore, an embedding coefficient k2 is introduced to characterize the influence of the pressure of the robotic arm and the material on the embedding depth, that is, the embedding depth of the abrasive grain is k2D, and 0 < k2 < 1. Its size is related to the pressure of the robotic arm and the material properties and can be obtained through parameter experiments; when rough materials come into contact, the actual contact area is less than the total area. Therefore, a contact coefficient k3 is introduced to characterize the contact error of the rough surface, that is, the actual contact area A = k3A0. Similarly, 0 < k3 < 1, and its size is related to the pressure of the robotic arm and the material properties and can be obtained through parameter experiments; in addition, when the pressure is small, it is considered that k2 and k3 change linearly with the change of pressure. Therefore, in particular, a six-axis force sensor is used in this invention to compensate for the pressure of the robotic arm, and the coefficients k2 and k3 change linearly according to the compensated pressure, making it more accurate.

[0068] Among them, the removal rate of a single microscopic abrasive grain is:

[0069]

[0070] In the formula, ΔS is the particle embedding area, unit mm 2 , δ wThe particle embedding depth is expressed in mm. Due to the flexibility of the polishing head, the particle embedding depth δ is... w Much smaller than D, simplified to:

[0071] Therefore, the macroscopic polishing material removal rate ρ is:

[0072]

[0073] In the formula, N A This represents the total number of abrasive grains, which is related to the concentration of the polishing slurry.

[0074] 103) Calculate the row spacing of the macroscopic polishing trajectory, and obtain the curvature-adaptive row spacing by controlling the residual height. Fit the freeform surface to be processed to a NURBS surface, and the rational fraction of the surface is:

[0075]

[0076] In the formula, u and v are parameter variables on the surface; P ij These are control points on the surface; i and j represent the number of meshes; ω ij These are the weight factors associated with the control vertex; p and q are the number of weight factors; N i，p (u), N j，q (v) are defined as nonrational B-spline basis functions on node vectors U and V, respectively.

[0077] Furthermore, the curvature data at any point can be obtained from a known freeform NURBS surface.

[0078] The contact area between the polishing tool head and the component is considered as an ellipse, and the polishing process of the polishing tool head polishing the variable curvature component is modeled as shown in the attached figure. Figure 3 As shown, a coordinate system is established for the profile of the polishing tool head. The row spacing between the two polishing paths is K, and the residual height in the middle is h. h represents the surface quality after machining and can be determined in conjunction with surface quality requirements.

[0079] The formula for an ellipse is:

[0080]

[0081] In the formula, a and b are ellipse parameters, that is, the geometric contour parameters of the polishing tool. Different polishing tool heads will have different contour parameter formulas.

[0082] The implicit formula for calculating line spacing is proposed as follows:

[0083]

[0084] For a plane, K = 2x and y = h; for a convex surface, K1 = k4 * K and k4 < 1; for a concave surface, K1 = k4 * K and k4 > 1; where the value of k4 depends on both the material removal rate and the curvature, and can be derived from experimental data.

[0085] Based on the line spacing with different curvatures mentioned above, a novel adaptive line spacing path planning method combining curvature is derived. The specific line spacing control model is attached. Figure 3 As shown.

[0086] By planning the path based on the macroscopic optimal row spacing and the process parameters based on the microscopic removal mechanism, high-precision freeform surfaces can be obtained through control from two aspects.

[0087] The pose control method for robotic arm polishing freeform surfaces has the following key points:

[0088] This paper proposes a motion control method for freeform surface trajectory planning using point cloud control for single-point motion or circular fitting motion. The motion of complex surfaces is decomposed into single-point motions, with the position of each point determined by contact pressure and NURBS curvature. A motion control method for freeform surface polishing is proposed, employing point cloud fitting of the surface. Based on the kinematics of a six-DOF robotic arm, specific polishing motions are achieved, including point cloud positional motion and optimal attitude control, to improve accuracy and efficiency.

[0089] Specifically, the origin of the end-effector coordinate system of the robotic arm is set to the center of the spherical polishing head. The point cloud position is calculated using the surface normal and the radius of the polishing head to compensate for overcompression, thus obtaining the coordinates of the center of the spherical polishing head of the robotic arm when polishing a certain point on the surface.

[0090]

[0091] In the formula, x0 and y0 are points on the freeform surface, R is the radius of the spherical polishing head that has been compensated for the compression of the polishing head, and y′(x0) is the derivative of the explicit expression, corresponding to the slope parameter of the freeform surface.

[0092] Two principles are proposed for attitude control: a set tilt angle to ensure the rotational linear velocity at the contact position, thereby improving polishing efficiency, and to avoid interference with the curved surface. Therefore, based on the curvature variation law of the spherical polishing head, 45° is selected as the optimal tilt angle. An optimized schematic diagram is attached. Figure 4 As shown.

[0093] Since Euler angles have a gimbal lock, quaternions are chosen to specifically control the posture of the robotic arm. The formula for controlling the robotic arm at 45° using quaternions is omitted.

[0094] Example

[0095] A specific embodiment is a planetary motion removal model for workpiece polishing by a robotic arm, based on the present invention, as shown in the attached figure. Figure 5 As shown, a coordinate system is established on the workpiece, and the distance from the abrasive grain to the center of the workpiece is r. p The angle between the workpiece and the x-axis is θ. The polishing trajectory of the robotic arm is a circular motion with an angular velocity of ω1. A coordinate system B is established for the workpiece, and the coordinates of points on this system are:

[0096]

[0097] This involves establishing a coordinate system A for the polishing disk, and then transforming (x1, y1) in coordinate system A to coordinate system B.

[0098]

[0099] Where a and b are the horizontal and vertical distances between the workpiece center and the grinding disc center, respectively, as shown in the attached figure. Figure 4 As shown. The relative motion velocity v1 of the robotic arm is then:

[0100]

[0101] The relative velocity v2 between the polishing abrasive grains and the robotic arm is:

[0102]

[0103] Therefore, substituting the material removal mechanism formula, we get:

[0104]

[0105] Specific embodiment 2 is based on the point cloud fitting calculation of a six-DOF robotic arm on a cylindrical non-spherical surface, which is an extension of the present invention. The complex surface is decomposed into a point cloud with normal directions, including pose parameters i, j, and k:

[0106] The formula for aspherical surfaces is:

[0107]

[0108] The derivative is:

[0109] y′(x)=(7635133935419849*x^3) / 295147905179352825856-(2*x) / ((7987*(1

[0110] -(628039819199249043*x^2) / 72057594037927936000)^(1 / 2)) / 1000

[0111] +7987 / 1000)+(3792922117627449*x^5) / 590295810358705651712

[0112] -(4716231184335101*x^7) / 18889465931478580854784

[0113] +(42556219845811905*x^9) / 9671406556917033397649408

[0114] -(17289912387886335*x^11) / 1237940039285380274899124224

[0115] -(5016154035944402106441*x^3) / (72057594037927936000000

[0116] *((7987*(1-(628039819199249043

[0117] *x^2) / 72057594037927936000)^(1 / 2)) / 1000+7987 / 1000)^2*(1

[0118] -(628039819199249043*x^2) / 72057594037927936000)^(1 / 2));

[0119] Find the perpendicular normal to any point on the aspherical surface:

[0120]

[0121] This yields the normal parameters i, j, and k of the aspherical surfaces x and y on the generatrix, which are used for the pose adjustment of the robotic arm. From the above equation, the tilt angle is: α = arctan(f′(x0)), therefore the position of the center of the spherical polishing head can be obtained.

[0122]

[0123] In the formula, R is the radius of the polishing head that compensates for overcompression. The curvature data is obtained by substituting the cylindrical aspherical surface into the NURBS surface formula, and the point cloud data is obtained according to the following formula:

[0124]

[0125] In the formula, given the residual height h, the planar machining line spacing K is obtained according to the tool parameters, and then the line spacing coefficient k4 is optimized according to the curvature data to obtain the optimized x and y positions of the point cloud. Since the relationship between curvature and line spacing coefficient k4 is too complicated, this invention proposes to approximate the two as a linear relationship.

[0126] Therefore, this embodiment proposes a point cloud generation logic block diagram for a search algorithm, as shown in the attached diagram. Figure 6 As shown, the requirement is that the distance between two points is K, and the point lies on the surface. The starting point is (x0, y0). The search strategy is to increase the value of x on the surface y(x) with an appropriate step size until the next point that meets the conditions is found. This is used to generate the point cloud of all cylindrical aspherical generatrices.

[0127] Although the present invention has been described in detail above with general descriptions and specific embodiments, modifications or improvements can be made to it, which will be obvious to those skilled in the art. Therefore, all such modifications or improvements made without departing from the spirit of the present invention fall within the scope of protection claimed by the present invention.

Claims

1. A curvature adaptive polishing method based on a robot arm, characterized by, This method is based on a freeform surface polishing system based on a robotic arm, including a six-degree-of-freedom robotic arm, an end-effector polishing head, a six-dimensional force sensor, a worktable, and a tooling fixture; wherein the end-effector polishing head is driven by a motor and fixed to the end of the six-degree-of-freedom robotic arm; the six-dimensional force sensor is installed at the bottom of the end-effector polishing head; the tooling fixture is fixed to the worktable by bonding or welding, and the worktable is located within the nominal working range of the robotic arm; The method includes the following steps: 1) A high-precision surface polishing method combining macroscopic trajectory planning and microscopic material removal for freeform surfaces: This includes a multi-coefficient polishing removal mechanism and trajectory planning for robotic arm adaptive freeform surface polishing based on NURBS surface curvature parameters; based on the Preston equation and the robotic arm's geometric motion model, a variable-coefficient polishing removal mechanism formula is proposed according to the abrasive embedding mechanism and the rough material contact mechanism; based on NURBS surface characteristics, an efficient and high-precision variable-coefficient polishing trajectory planning method for robotic arm polishing of freeform surfaces is proposed, and the adaptive machining trajectory spacing is planned according to the curvature change of the freeform surface; the specific implementation method is as follows: 101) The relative motion of the robotic arm to the workpiece during polishing is proposed, and the relative motion speed between the abrasive grains and the workpiece is calculated based on the relative motion speed between the robotic arm and the workpiece; since there is also relative motion between the abrasive grains and the robotic arm, a speed coefficient is introduced. To characterize the relative motion speed between the polishing abrasive grains and the robotic arm, i.e. ,in The relative speed between the robotic arm and the workpiece. The relative velocity between the abrasive grains and the workpiece, and That is, the relative speed between the polishing abrasive grains and the workpiece is less than the relative speed between the robotic arm and the workpiece; 102) Based on the micro-polishing abrasive grain embedding mechanism and the macro-rough material contact mechanism, a Preston material removal equation is proposed; The depth to which the abrasive grains embed into the workpiece is determined by the pressure of the robotic arm and the inherent properties of the material, and the size is smaller than the diameter D of the abrasive grains; therefore, an embedding coefficient is introduced. To characterize the pressure of the robotic arm and the effect of the material on the embedding depth, i.e., the abrasive embedding depth is... ,and The size is related to the pressure of the robotic arm and the material properties, and is obtained through parameter experiments; when rough materials come into contact, the actual contact area will be smaller than the total area, therefore a contact coefficient is introduced. To characterize the contact error of a rough surface, i.e., the actual contact area. Similarly, Additionally, when stress is low, it is believed that... and The force changes linearly with pressure; a six-dimensional force sensor is used to compensate for the pressure on the robotic arm, and the coefficient... and Based on the linear change of the compensated pressure; The removal rate of individual micro-abrasive particles is: In the formula, For particle embedding area, in units , The depth of particle embedding is expressed in mm; due to the flexibility of the polishing head, the depth of particle embedding into the component is... Much smaller than D, simplified to: ; Therefore, the macroscopic polishing material removal rate for: In the formula, This represents the total number of abrasive grains, which is related to the concentration of the polishing slurry. 103) Calculate the row spacing of the macroscopic polishing trajectory, and obtain the curvature-adaptive row spacing by controlling the residual height; fit the freeform surface to be processed into a NURBS surface, and the rational fraction of the surface is: In the formula, u and v are parameter variables on the surface; These are control points on the surface; i and j represent the number of meshes. q are the weight factors associated with the control vertex; p and q are the number of weight factors. These are defined as nonrational B-spline basis functions on the node vectors U and V, respectively. Furthermore, given a freeform NURBS surface, the curvature data at any point can be obtained; The contact area between the polishing tool head and the component is considered as an ellipse. The polishing process of the polishing tool head polishing the variable curvature component is modeled. A coordinate system is established for the contour of the polishing tool head. The row spacing between the two polishing paths is K. Then the residual height in the middle is h, and h represents the surface quality after processing. The formula for an ellipse is: In the formula, a and These are ellipse parameters, i.e., the geometric contour parameters of the polishing tool. Different polishing tool heads will have different contour parameter formulas. The implicit formula for calculating line spacing is proposed as follows: For a plane, when K = 2x, y = h; for a convex surface, then... = k4*K, and k4<1; for concave surfaces, then = k4*K, and k4>1; Based on the line spacing with different curvatures, an adaptive line spacing path planning that combines curvature is obtained; The path is planned based on the macroscopic optimal row spacing, and the process parameters are planned based on the microscopic removal mechanism. By controlling these two aspects, a high-precision freeform surface can be obtained. 2) Pose control method for freeform surface polishing by robotic arm: Motion control for freeform surface polishing is achieved by using point cloud fitting to the surface. The specific polishing motion is realized based on the kinematics of a six-degree-of-freedom robotic arm, including point cloud position motion and optimal posture control.

2. The curvature adaptive polishing method based on a robotic arm according to claim 1, characterized in that, In step 2), the specific implementation method is as follows: The trajectory planning of freeform surfaces adopts a single-point motion mode controlled by point cloud or a circular fitting motion mode, decomposing the motion of complex surfaces into single-point motions, with the position of each point determined by contact pressure and NURBS curvature. A motion control method for freeform surface polishing is proposed by using point cloud fitting of the surface, and the specific polishing motion is realized based on the kinematics of a six-degree-of-freedom robotic arm, including point cloud position motion and optimal attitude control, to improve accuracy and efficiency.

3. The curvature adaptive polishing method based on a robotic arm according to claim 2, characterized in that, The origin of the end-effector coordinate system of the robotic arm is set to the center of the spherical polishing head. The point cloud position is calculated using the surface normal and the radius of the polishing head to compensate for overcompression, thus obtaining the coordinates of the center of the spherical polishing head of the robotic arm when polishing a certain point on the surface: In the formula, , Let be a point on the free surface. To compensate for excessive compression of the polishing head, the radius of the spherical polishing head is adjusted. The derivative of the explicit expression corresponds to the slope parameter of the freeform surface; Two principles are proposed for attitude control: a set tilt angle to ensure the self-rotation linear velocity at the contact position to improve polishing efficiency and to avoid interference with the curved surface. Therefore, based on the curvature variation law of the spherical polishing head, 45° is selected as the optimal tilt angle. Among them, because Euler angles have a gimbal lock, quaternions are chosen to specifically control the posture of the robotic arm.

4. The curvature adaptive polishing method based on a robotic arm according to claim 1, characterized in that, The polishing path motion of the free-form surface is completed by programming the six-degree-of-freedom robotic arm. When the worktable is in the middle of the motion range, the whole system is in the optimal working state, with the best vibration state and motion accuracy. At the same time, based on the real-time feedback of three-dimensional force and three-dimensional torque by the six-dimensional force sensor and the mechanical properties of the polishing head, a repeatability positioning accuracy of 0.02mm is achieved.

5. The curvature adaptive polishing method based on a robotic arm according to claim 1, characterized in that, The six-dimensional force sensor is positioned at the end of the polishing tool head to collect the normal polishing force of the polishing head. The data collected by the six-dimensional force sensor is fed back to the compensating motion of the robotic arm, thereby forming a closed-loop control to compensate for the error of the polishing force.

6. The curvature adaptive polishing method based on a robotic arm according to claim 5, characterized in that, The polishing force at the end of the robotic arm is collected by a six-dimensional force sensor. Based on the force-displacement compression performance of the polishing head, the displacement control perpendicular to the processing surface at the end of the robotic arm is fed back in real time. This compensates for the error in the pressure control of the robotic arm, thereby compensating for the influence of polishing pressure on the indentation of polishing particles and the actual contact area, i.e., compensating for the embedding coefficient. and contact coefficient When the pressure is low, it is assumed that the embedding coefficient and contact coefficient change linearly with the polishing pressure.

7. The curvature adaptive polishing method based on a robotic arm according to claim 1, characterized in that, The tooling fixture includes a support base plate and an upper slotted fixture; the support base plate has air suction holes, and the workpiece is fixed by vacuum suction and bonding. One side of the upper slotted fixture is adjustable, and the other side is fixed.

8. The curvature adaptive polishing method based on a robotic arm according to claim 7, characterized in that, The support base plate is equipped with a ball screw, and a movable baffle is threaded onto the ball screw. The linear motion of the movable baffle can be achieved by the rotation of the ball screw.

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