Magnetorheological magic angle polishing tool and method

Abstract

A magnetorheological magic angle polishing tool and processing method are disclosed. The tool includes a polishing wheel, a drive motor, a magnet, and a magnetorheological fluid system, and integrates a magic angle control rotary table (C-axis) driven by the motor. The polishing wheel, its drive motor, and the magnet are all mounted on this rotary table, forming a tool head that can rotate as a whole. By adding this independent C-axis, this invention achieves decoupling of processing path planning and magic angle control. The control software calculates and controls the C-axis rotation in real time based on the workpiece surface equation, the planned path, and the calibrated optimal magic angle, so that the polishing wheel can maintain a stable magic angle with the local normal of the workpiece at any point during the processing of complex curved surfaces in a plane perpendicular to the feed direction. This method fundamentally overcomes the problems of angle instability and path limitation caused by traditional reliance on machine tool XY axis linkage, and significantly improves the mid-frequency error suppression effect, process adaptability, and shaping capability.

Description

Technical Field

[0001] This invention relates to the field of optical precision manufacturing equipment technology, specifically a magnetorheological magic angle polishing tool and processing method. Background Technology

[0002] Magnetorheological finishing (MRF) technology uses a controllable magnetic field to create a flexible "polishing ribbon" of a magnetorheological fluid with rheological properties in the polishing area, achieving nanoscale precision removal of workpiece material. It has demonstrated excellent performance in correcting low-frequency surface shape errors. However, during the polishing process, the relative motion between the tool and the workpiece can easily introduce periodic texture or ripple errors within a specific spatial frequency range (i.e., the mid-frequency band). These errors can severely affect the imaging quality of optical systems, especially in high-power laser systems and extreme ultraviolet lithography.

[0003] Recent research has discovered a specific process angle in magnetorheological polishing that is closely related to the removal function characteristics; this is known as the "magic angle." Polishing at this angle minimizes the transfer function of the process system to intermediate frequency (IF) errors, effectively suppressing or eliminating IF ripple generation without affecting the convergence of errors in other frequency bands. This discovery provides a new approach to fundamentally solve the IF problem in magnetorheological polishing.

[0004] However, in practical engineering applications, especially when machining complex surfaces such as aspherical and free-form surfaces, achieving and maintaining this "magic angle" stably and accurately faces significant challenges. Traditional methods lack dedicated angle control mechanisms and typically rely on the coordinated movement of the X and Y axes of the CNC machine tool, approximating the relative angle between the tool and workpiece by changing the direction of the polishing path in the horizontal plane. This method has significant limitations: First, for steep surfaces with drastic curvature changes, path planning in the XY plane alone cannot maintain a constant optimal magic angle between the tool and workpiece local normals in real time at each point of the machining path, leading to decreased angle control accuracy and weakening or even eliminating the ripple suppression effect of the magic angle. Second, this method strongly couples the selection of the machining path with angle control, forcing the path direction to be limited to a few fixed directions (such as orthogonal grating paths). It cannot flexibly and adaptively plan the optimal polishing path according to the actual spatial distribution characteristics of the workpiece surface shape error, severely restricting the improvement of the process's shaping capability and machining efficiency. Furthermore, relying on high-frequency, small-range linkage of the machine tool's XY axes to simulate angle changes places higher demands on the machine tool's dynamic response performance and motion accuracy, increasing the complexity of process control.

[0005] Therefore, there is an urgent need for a dedicated tool and method that can decouple polishing path planning from magic angle control, enabling the tool to independently, in real time, and precisely adjust the posture of the polishing tool at every local position on the workpiece surface along any planned path. This ensures the stable realization of the magic angle throughout the entire processing, especially on complex curved surfaces, thereby fully leveraging the potential of the magic angle process to obtain higher precision optical surfaces without mid-frequency ripples. Summary of the Invention

[0006] To address the problems of unstable angle control and limited path flexibility on complex curved surfaces caused by the reliance on machine tool XY axis linkage in existing magnetorheological polishing technology to achieve the "magic angle" effect, this invention provides a magnetorheological magic angle polishing tool and processing method. By integrating an independent rotational degree of freedom into the tool head, the polishing path control and magic angle control are completely decoupled. This allows for the dynamic, precise, and stable maintenance of the optimal polishing angle on any complex curved surface and any planned path, significantly improving the mid-frequency error suppression effect and process adaptability.

[0007] The technical solution of the present invention is as follows: A magnetorheological magic angle polishing tool, characterized by comprising: The spindle connection part has a rotary table fixing plate for rigid connection with the end of the machine tool spindle; The C-axis rotation drive unit includes a magic angle rotary table motor fixed to the rotary table mounting plate, and a magic angle control rotary table driven by the magic angle rotary table motor. The magic angle control rotary table can rotate relative to the rotary table mounting plate about a C-axis parallel to the machine tool spindle axis. The polishing function unit is fixedly mounted on the magic angle control rotary table and rotates with it around the C-axis. The polishing function unit includes: A polishing wheel assembly includes a polishing wheel and a polishing wheel rotary motor for driving the polishing wheel to rotate about its own axis, wherein the transmission mechanism between the polishing wheel rotary motor and the polishing wheel is disposed within the polishing functional unit; The magnetic field generating assembly includes a magnet disposed on one side of the polishing wheel and a magnet fixing bracket for mounting and adjusting the magnet; The fluid management assembly includes a magnetorheological fluid nozzle for supplying magnetorheological fluid between the polishing wheel and the workpiece, and a magnetorheological fluid recycler for recycling the used magnetorheological fluid. The polishing function unit is driven to rotate around the C-axis by the C-axis rotation drive unit, which enables the polishing wheel to independently and dynamically adjust its circumferential orientation at the machining point while the machine tool is performing machining movements on the X, Y, Z, A, and B axes.

[0008] Furthermore, the polishing wheel assembly also includes a polishing wheel bracket, on which the polishing wheel is mounted via an angular contact bearing; the polishing wheel rotation motor drives the polishing wheel via a synchronous belt drive mechanism, which includes an output synchronous belt pulley on the motor output shaft, a driven synchronous belt pulley on the polishing wheel shaft, and a synchronous belt connecting the two.

[0009] Furthermore, the magnet fixing bracket in the magnetic field generating assembly is provided with a magnet adjusting screw for finely adjusting the working gap between the pole face of the magnet and the surface of the polishing wheel.

[0010] Furthermore, the magnetorheological fluid nozzle in the fluid management component is mounted via a nozzle bracket that can slide and lock along a preset U-shaped groove to adjust its outlet position; the recovery port of the magnetorheological fluid recoverer is arranged around the outer periphery of the polishing wheel portion, and the gap between the recovery port and the surface of the polishing wheel is adjustable.

[0011] Furthermore, the rotary table with magic angle control allows the polishing wheel to rotate by an angle dynamically adjusted according to the actual workpiece normal during the machining process via the rotary table's motor, achieving higher quality mid-frequency error control.

[0012] Second, the present invention also provides a processing method using the above-mentioned magnetorheological angle polishing tool, characterized by comprising the following steps: S1: Determine the removal function R(x,y) of the polishing tool under specific process parameters, and calibrate the magic angle that can suppress mid-frequency errors based on the frequency characteristics of the removal function. i 魔法 ; S2: Obtain the surface equation z=f(x,y) of the workpiece to be processed, and independently of the angle control requirements, plan the processing path point set {P} based on the shaping target. i | P i = }; S3: For each point P on the path i Perform dynamic angle planning: a) Calculate the unit normal vector of the workpiece surface at this point based on the surface equation. ; b) Based on the machine tool kinematics model, calculate the required A-axis rotation angle of the machine tool to align or parallel the normal to the preset machining direction at this point. and B-axis rotation angle ; c) Real-time route planning Determine the unit tangent vector of the feed direction at that point. ; d) Based on the calibrated magic angle i魔法 The calculated normal vector tangent vector By solving spatial geometry, the required compensation angle for the C-axis of the polishing tool is determined. This enables the execution of the A and B axes of the machine tool. After rotation, it is then executed via the C-axis. Rotation ensures that the effective working surface of the polishing wheel is perpendicular to the instantaneous feed direction to the local surface of the workpiece. In the plane, the angle between the projections of the normals of the two is always . i 魔法 ; S4: Based on the surface error distribution and removal function, calculate the dwell time at each point on the path and convert it into the feed rate V. i ; S5: Integrated Path Coordinates Machine axis rotation commands, tool axis rotation commands, and feed rate V i Generate CNC programs; S6: Execute machining. The machine tool axes move according to the program to control the workpiece position and tool path. At the same time, the C-axis of the polishing tool rotates in real time according to the program instructions to ensure that the magic angle is achieved at all points on the path. i 魔法 The stability of magic is maintained.

[0013] Furthermore, the planning of the processing path in step S2 can adopt grating lines, spiral lines, concentric circles, or free paths that are completely adaptive according to the direction of the surface error gradient. The selection is not constrained by the tool-workpiece relative angle relationship required to achieve the magic angle.

[0014] Compared with the prior art, the beneficial effects of the present invention are as follows: 1) A "magic angle control rotary table" (C-axis) directly driven by a motor is integrated into the traditional magnetorheological polishing tool head, giving the tool itself a degree of freedom to rotate around the spindle. This is specifically used to precisely control the deflection angle of the polishing wheel relative to the local normal of the workpiece, thereby completely separating the "angle control" function from the machine tool's "path motion" (XY axis linkage).

[0015] 2) By achieving dynamic and precise maintenance of the magic angle on complex curved surfaces (especially steep aspherical surfaces), the problem of weakened process effect caused by angle drift in traditional methods is fundamentally solved, making the acquisition of ripple-free surfaces more stable and reliable, and greatly improving the convergence level of the mid-frequency band.

[0016] 3) The tool can adapt to machining paths with arbitrary complex surfaces and arbitrary orientations. The liberation of path planning allows process engineers to design paths in a targeted manner to more efficiently correct errors at specific spatial frequencies, improving the flexibility and deterministic shaping capabilities of the entire process.

[0017] 4) Flexible path planning capabilities combined with stable magic angle effects can optimize material removal strategies while ensuring ultra-smooth surface quality (suppressing ripples), potentially shortening shaping time and achieving a dual improvement in efficiency and precision. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the magnetorheological magic angle polishing tool of the present invention, wherein a is a frontal half-sectional view of the polishing tool and b is a side view of the polishing tool.

[0019] Figure 2 This is a side view of the magnetorheological magic angle polishing tool of the present invention. In the diagram: 1-Magic Angle Rotary Table Rotary Motor, 2-Rotary Table Fixing Plate, 3-Magic Angle Control Rotary Table, 4-Polishing Wheel Bracket Fixing Flange, 5-Magnetorheological Fluid Nozzle, 6-Output Synchronous Belt Pulley, 7-Magnetorheological Fluid Recoverer, 8-Synchronous Belt, 9-Angular Contact Bearing, 10-Passive Synchronous Belt Pulley, 11-Polishing Wheel Bracket, 12-Bearing Spacer Ring, 13-Nozzle Position Fixing U-Shaped Groove, 14-Magnet Fixing Bracket, 15-Polishing Wheel, 16-Magnet, 17-Magnet Adjusting Screw, 18-Recoverer Bracket, 19-Nozzle Bracket, 20-Polishing Wheel Rotary Motor, 21-Bearing End Cap. Detailed Implementation

[0020] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings. The following examples are only used to more clearly illustrate the technical solutions of the present invention and should not be construed as limiting the scope of protection of the present invention.

[0021] Please see Figure 1 and Figure 2 As shown in the figure, this embodiment of the magnetorheological magic angle polishing tool integrates a C-axis that can be precisely controlled by software, enabling dynamic and independent adjustment of the polishing tool head orientation. The tool includes: The spindle connection part has a rotary table fixing plate 2 for rigid connection with the end of the machine tool spindle; The C-axis rotation drive unit includes a magic angle rotary table motor 1 fixed on the rotary table mounting plate 2, and a magic angle control rotary table 3 driven by the magic angle rotary table motor 1. The magic angle control rotary table 3 can rotate relative to the rotary table mounting plate 2 about a C-axis parallel to the machine tool spindle axis. A polishing function unit is fixedly mounted on the magic angle control rotary table 3 and rotates with it around the C-axis. This polishing function unit includes: A polishing wheel assembly includes a polishing wheel 15 and a polishing wheel rotary motor 20 for driving the polishing wheel 15 to rotate about its own axis. The transmission mechanism between the polishing wheel rotary motor 20 and the polishing wheel 15 is disposed within the polishing functional unit. The polishing wheel assembly also includes a polishing wheel bracket 11, on which the polishing wheel 15 is mounted via an angular contact bearing 9. The polishing wheel rotary motor 20 drives the polishing wheel 15 via a synchronous belt drive mechanism, which includes an output synchronous belt pulley 6 disposed on the motor output shaft, a passive synchronous belt pulley 10 disposed on the polishing wheel shaft, and a synchronous belt 8 connecting the two.

[0022] The magnetic field generating assembly includes a magnet 16 disposed on one side of the polishing wheel 15 and a magnet fixing bracket 14 for mounting and adjusting the magnet 16; the magnet fixing bracket 14 is provided with a magnet adjusting screw 17 for finely adjusting the working gap between the pole face of the magnet 16 and the surface of the polishing wheel 15.

[0023] The fluid management assembly includes a magnetorheological fluid nozzle 5 for supplying magnetorheological fluid between the polishing wheel 15 and the workpiece, and a magnetorheological fluid recycler 7 for recycling the used magnetorheological fluid; the magnetorheological fluid nozzle 5 is mounted via a nozzle bracket 19 that can slide and lock along a preset U-shaped groove 13 to adjust its outlet position; the recycling port of the magnetorheological fluid recycler 7 is arranged around the outer periphery of a portion of the polishing wheel 15, and the gap between the recycling port and the surface of the polishing wheel is adjustable.

[0024] The polishing function unit is driven to rotate around the C-axis by the C-axis rotation drive unit, which enables the polishing wheel 15 to independently and dynamically adjust its circumferential orientation at the machining point while the machine tool performs machining movements on the X, Y, Z, A, and B axes.

[0025] The Magic Angle Control Rotary Table 3 is connected to the spindle of the five-axis CNC machine tool via a rotary table mounting plate 2, serving as the overall base of the tool. The Magic Angle Rotary Table rotary motor 1 is fixed to the rotary table mounting plate 2, and its output shaft is connected to the input shaft of the Magic Angle Control Rotary Table 3 via a coupling (not shown in the figure), driving the entire control rotary table 3 to rotate around the C-axis, which is perpendicular to the machine tool spindle axis. The polishing wheel bracket 11 is fixed to the Magic Angle Control Rotary Table 3. The polishing wheel 15 is supported within the polishing wheel bracket 11 by a pair of high-precision angular contact bearings 9, and is axially positioned and pre-tightened by bearing end caps 21 and bearing spacers 12. A passive synchronous pulley 10 is mounted on the shaft end of the polishing wheel 15. The polishing wheel rotary motor 20 is fixed to the Magic Angle Control Rotary Table 3, and its output shaft is mounted with an output synchronous pulley 6. A synchronous belt 8 connects the output synchronous pulley 6 and the passive synchronous pulley 10, thereby transmitting power to the polishing wheel 15, causing it to rotate at high speed around its own axis. Magnet 16 is mounted on polishing wheel bracket 11 via magnet fixing bracket 14, located near the working area of ​​polishing wheel 15. By turning magnet adjusting screw 17, the gap between the bottom of magnet 16 and the surface of polishing wheel 15 can be finely adjusted, thereby precisely controlling the magnetic field strength acting on the magnetorheological polishing zone. Magnetorheological fluid nozzle 5 is mounted via nozzle bracket 19 and embedded in nozzle position fixing U-shaped groove 13. By adjusting its position in the U-shaped groove and locking it, the position of magnetorheological fluid sprayed onto the surface of polishing wheel 15 can be precisely controlled. Magnetorheological fluid collector 7 is mounted via collector bracket 18, with its collection port maintaining a gap of approximately 0.1-0.2 mm from the surface of polishing wheel 15, ensuring that the used magnetorheological fluid can be efficiently recovered and recycled.

[0026] By controlling the C-axis rotation angle of the rotary table 3 in real time using the magic angle control, the relative angle between the polishing line of the polishing wheel 15 and the local surface of the workpiece can be stably maintained at a predetermined "magic angle" under any processing path and curved surface posture. i 魔法 This achieves full-area polishing without mid-frequency ripples. The specific control steps are as follows: S1: Determine the removal function R(x,y) of the polishing tool under specific process parameters (magnetic field strength, polishing wheel speed, abrasive concentration, etc.), and calibrate the magic angle that can suppress mid-frequency errors based on the frequency characteristics of the removal function. i 魔法 ; S2: Input the surface equation z=f(x,y) of the aspherical component to be processed. Based on the set path step d, plan a continuous processing path on the workpiece surface and generate a discretized path point set {P}. i | P i = }; S3: For each point P on the path iPerform dynamic angle planning: S31. Calculate the unit normal vector of the workpiece surface at this point based on the surface equation. ; S32. For each control point on the path ( The surface normal vector at that point is calculated based on the surface equation z = f(x, y). The CNC machine tool needs to use this normal vector, through the linkage of its A-axis and B-axis, to tilt the workpiece to a suitable orientation, so that the local surface normal direction at that point is basically aligned with (or at a predetermined machining angle) the tool spindle direction (Z-axis). The rotation amounts of the A and B axes... and The slope of the surface at that point is determined by the following formula: S33. For each control point, calculate the instantaneous tangent direction of the path at that point based on the preceding and following path points, i.e., the path travel direction vector: S34. The workpiece has been tilted to the position on machine tool axes A and B. and Then, to ensure that the angle between the polishing line of the polishing wheel 15 and the surface normal of the local processing point is accurate, i 魔法 Compensation rotation must be achieved via the C-axis of the tool head. The C-axis at point... The required compensation angle ∠c Calculated using the following formula: In the formula, φi is the angle between the path direction and the x-axis.

[0027] S4: Based on the surface error distribution and removal function, calculate the dwell time at each point on the path and convert it into the feed rate V. i ; S5: Integrated Path Coordinates Machine axis rotation commands, tool axis rotation commands, and feed rate V i Generate CNC programs; S6: Execute machining. The machine tool axes move according to the program to control the workpiece position and tool path. At the same time, the C-axis of the polishing tool rotates in real time according to the program instructions to ensure that the magic angle is achieved at all points on the path. i 魔法 The stability of magic is maintained.

[0028] This embodiment calculates the angle matching the removal function using software. A rotary table motor controls the entire polishing wheel to rotate in real-time to the magic angle state. The polishing wheel is mounted on a polishing wheel bracket by angular contact bearings and secured by bearing blocks. A passive synchronous pulley is fixed to the polishing wheel shaft end, and the polishing wheel rotary motor drives the passive synchronous pulley to rotate, thus rotating the polishing wheel. The magnet is fixed to the bracket with bolts, and the gap between the bottom of the magnet and the polishing wheel can be adjusted by adjusting screws to control the magnetic field strength on the polishing wheel surface. Magnetorheological fluid flows out through a nozzle, and the position of the magnetorheological fluid nozzle can be adjusted and fixed by a U-shaped groove for nozzle positioning. The magnetorheological fluid is recovered by a magnetorheological fluid collector, which maintains a gap of approximately 0.1 to 0.2 mm with the polishing wheel to ensure complete recovery. Based on the processing path and orientation, the c-axis rotation at each position is calculated. During the processing, the angle between the tool and the path direction at each point is controlled in real-time to match the removal function, thereby ensuring a ripple-free processing cycle effect under various processing path conditions.

[0029] The above description is merely a preferred embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural or procedural modifications made based on the description and drawings of the present invention, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of the present invention.

Claims

1. A magnetorheological magic angle polishing tool, characterized in that, include: The spindle connection part has a rotary table fixing plate (2) for rigid connection with the end of the machine tool spindle. The C-axis rotation drive unit includes a magic angle rotary table motor (1) fixed on the rotary table mounting plate (2) and a magic angle control rotary table (3) driven by the magic angle rotary table motor (1). The magic angle control rotary table (3) can rotate relative to the rotary table mounting plate (2) about a C-axis parallel to the machine tool spindle axis. A polishing function unit is fixedly mounted on the magic angle control rotary table (3) and rotates with it around the C-axis. The polishing function unit includes: The polishing wheel assembly includes a polishing wheel (15) and a polishing wheel rotary motor (20) for driving the polishing wheel (15) to rotate about its own axis. The transmission mechanism between the polishing wheel rotary motor (20) and the polishing wheel (15) is provided in the polishing functional unit. The magnetic field generating assembly includes a magnet (16) disposed on one side of the polishing wheel (15) and a magnet fixing bracket (14) for mounting and adjusting the magnet (16). The fluid management assembly includes a magnetorheological fluid nozzle (5) for supplying magnetorheological fluid between the polishing wheel (15) and the workpiece, and a magnetorheological fluid recycler (7) for recycling the used magnetorheological fluid. The polishing function unit is driven to rotate around the C-axis by the C-axis rotation drive unit, which can independently and dynamically adjust the circumferential orientation of the polishing wheel (15) at the processing point while the machine tool is performing processing movements on the X, Y, Z, A, and B axes.

2. The magnetorheological magic angle polishing tool according to claim 1, characterized in that, The polishing wheel assembly also includes a polishing wheel bracket (11), on which the polishing wheel (15) is mounted via an angular contact bearing (9); the polishing wheel rotary motor (20) drives the polishing wheel (15) via a synchronous belt drive mechanism, which includes an output synchronous belt pulley (6) on the motor output shaft, a passive synchronous belt pulley (10) on the polishing wheel shaft, and a synchronous belt (8) connecting the two.

3. The magnetorheological magic angle polishing tool according to claim 1, characterized in that, The magnet fixing bracket (14) in the magnetic field generating assembly is provided with a magnet adjusting screw (17) for finely adjusting the working gap between the pole face of the magnet (16) and the surface of the polishing wheel (15).

4. The magnetorheological magic angle polishing tool according to claim 1, characterized in that, The magnetorheological fluid nozzle (5) in the fluid management assembly is mounted via a nozzle bracket (19) that can slide and lock along a preset U-shaped groove (13) to adjust its liquid outlet position; the recovery port of the magnetorheological fluid recoverer (7) is arranged around the outer periphery of the polishing wheel (15), and the gap between the recovery port and the surface of the polishing wheel is adjustable.

5. The magnetorheological magic angle polishing tool according to claim 1, characterized in that... With a magic angle control rotary table, the polishing wheel can be rotated by an angle by controlling the actual workpiece normal during the processing through the magic angle rotary table rotary motor (1), which can achieve higher quality medium frequency error control.

6. A processing method using a magnetorheological magic angle polishing tool as described in any one of claims 1-5, characterized in that, Includes the following steps: S1: Determine the removal function R(x,y) of the polishing tool under specific process parameters, and calibrate the magic angle that can suppress mid-frequency errors based on the frequency characteristics of the removal function. θ 魔法 ; S2: Obtain the surface equation z=f(x,y) of the workpiece to be processed, and independently of angle control requirements, plan the processing path point set {P} based on the shaping target. i | P i = }; S3: For each point P on the path i Perform dynamic angle planning: a) Calculate the unit normal vector of the workpiece surface at this point based on the surface equation. ; b) Based on the machine tool kinematics model, calculate the required A-axis rotation angle of the machine tool to align or parallel the normal to the preset machining direction at this point. and B-axis rotation angle ; c) Real-time route planning Determine the unit tangent vector of the feed direction at that point. ; d) Based on the calibrated magic angle θ 魔法 The calculated normal vector tangent vector By solving spatial geometry, the required compensation angle for the C-axis of the polishing tool is determined. This enables the execution of the A and B axes of the machine tool. After rotation, it is then executed via the C-axis. Rotation ensures that the effective working surface of the polishing wheel (15) is perpendicular to the instantaneous feed direction to the local surface of the workpiece. In the plane, the angle between the projections of the normals of the two is always . θ 魔法 ; S4: Based on the surface error distribution and removal function, calculate the dwell time at each point on the path and convert it into the feed rate V. i ; S5: Integrated Path Coordinates Machine axis rotation commands, tool axis rotation commands, and feed rate V i Generate CNC programs; S6: Execute machining. The machine tool axes move according to the program to control the workpiece position and tool path. At the same time, the C-axis of the polishing tool rotates in real time according to the program instructions to ensure that the magic angle is achieved at all points on the path. θ 魔法 The stability of magic is maintained.

7. The processing method according to claim 6, characterized in that, The planning of the machining path in step S2 can adopt grating lines, spiral lines, concentric circles, or free paths that are completely adaptive according to the direction of the surface error gradient. The selection is not constrained by the tool-workpiece relative angle relationship required to achieve the magic angle.

Images