Calculation Method for Posture of Integrated Machining Position of Magnetorheological Finishing Multi-Faceted Co-Body Components

Abstract

The present invention relates to the field of optical processing technology, and particularly to a method for calculating the machining pose of a magnetorheological polishing multi-faceted coaxial element. According to the surfaces of each mirror body in the multi-faceted coaxial element to be machined, the corresponding trajectory segments of each mirror body during machining are determined; by combining the corresponding trajectory segments of each mirror body, the ideal polishing trajectory during machining is obtained; the polishing gap of the machining tool during machining is set, and the normal vectors of each point of the mirror body in the multi-faceted coaxial element are determined; according to the polishing gap and the normal vectors of each point, the actual polishing trajectory of the machining tool and the machining pose of the machining tool are obtained, and the machining pose of the machining tool is derived. The present invention realizes the machining pose control of coaxial multi-faceted coaxial elements, and reduces the redundant manufacturing time of multi-stage machining.

Description

Technical Field

[0001] The present invention belongs to the technical field of optical processing, and particularly relates to a method for calculating the pose of integrated machining of a multi-faceted co-body element by magnetorheological polishing. Background Art

[0002] Nowadays, most high-performance imaging optical systems in the world adopt multi-mirror reflective optical systems, which can be divided into coaxial systems and off-axis systems. Common ones are off-axis three-mirror and coaxial three-mirror types. However, the alignment and adjustment of single-element systems are difficult and costly, and the volume and mass of opto-mechanical structural parts are large. Therefore, a design scheme of multi-faceted co-integration has emerged. The integration of the primary and secondary mirrors of off-axis three-mirror can be seen in the paper "Design of an Off-Axis Three-Mirror Optical System with Integrated Primary and Secondary Mirrors" published in Infrared and Laser Engineering. The integration of the primary and secondary mirrors of coaxial three-mirror can be seen in the paper "Fabrication of the LSST Monolithic Primary-Tertiary Mirror" published in Modern Technologies in Space- and Ground-based Telescopes and Instrumentation II. In this paper, the multi-faceted co-body surface is polished using a stress disk tool, and its removal function size is relatively large and has no obvious directionality. Therefore, there is more flexibility in the process to control.

[0003] In the shaping stage of large-aperture optical elements, the magnetorheological polishing scanning removal processing method is becoming more and more mainstream, and the magnetorheological polishing removal function is relatively small and directional. The processing mode of magnetorheological polishing needs to evaluate the processing cycle. The off-axis multi-faceted co-body mirror can adopt two processes to scan and process the main mirror and the secondary mirror respectively, without generating too much redundant manufacturing time; while for the coaxial multi-faceted co-body mirror, if the grating trajectory is still used and the main mirror and the secondary mirror are scanned and processed in two processes respectively, during the scanning process of the main mirror, there will be an idle running time between the central hole and the secondary mirror, which will cause too much redundant manufacturing time; if the main mirror and the secondary mirror are regarded as one process, it is necessary to accurately control the pose change at the joint of the main mirror and the secondary mirror, thereby increasing the difficulty of program development; if a circular trajectory is adopted, if the workpiece rotates, the design of the mirror body support and the rotating mechanical structure is complex, increasing the additional design cost. If the tool rotates, the machine tool, as the carrier of the magnetorheological tool, has a high degree of freedom requirement, thus requiring a high-precision calculation method and a high-performance multi-degree-of-freedom machine tool for support. Summary of the Invention

[0004] In view of this, the present invention aims to provide a method for calculating the machining position and attitude of a multi-faceted coaxial element in integrated magnetorheological polishing, obtaining the trajectory segments of the surfaces of each mirror body in the multi-faceted coaxial element to be machined, and then determining the total polishing trajectory during machining; according to the set polishing gap and the normal vectors of each point of the mirror body in the multi-faceted coaxial element, obtaining the machining position and machining attitude of the machining tool, and combining the total polishing trajectory, the machining position of the machining tool, and the machining attitude, realizing the integrated machining position and attitude control of the coaxial multi-faceted element, and reducing the redundant manufacturing time of multi-stage machining.

[0005] To achieve the above object, the technical solution of the present invention is realized as follows:

[0006] A method for calculating the machining position and attitude of a multi-faceted coaxial element in integrated magnetorheological polishing, comprising:

[0007] S1: Determine the trajectory segments corresponding to each mirror body during machining according to the surfaces of each mirror body in the multi-faceted coaxial element to be machined; combine the trajectory segments corresponding to each mirror body to obtain the ideal polishing trajectory during machining;

[0008] S2: Set the polishing gap during machining of the machining tool and determine the normal vectors of each point of the mirror body in the multi-faceted coaxial element;

[0009] S3: According to the ideal polishing trajectory obtained in step S1, and the polishing gap and the normal vectors of each point obtained in step S2, obtain the actual polishing trajectory of the machining tool and the machining attitude of the machining tool;

[0010] S4: Combine the actual polishing trajectory and the machining attitude obtained in step S3 to obtain the machining position and attitude of the machining tool.

[0011] Further, step S1 includes:

[0012] S11: Obtain the surface equations of the main mirror body and the secondary mirror body in the multi-faceted coaxial element;

[0013] S12: Generate the first trajectory segment with one single-side ring arm of the main mirror body as a reference;

[0014] S13: Generate the second trajectory segment with the remaining ring arms of the main mirror body except for the ring arm in S12 as a reference;

[0015] S14: Generate the third trajectory segment with the secondary mirror body as a reference;

[0016] S15: According to the first trajectory segment, the second trajectory segment, and the third trajectory segment obtained in steps S12 to S14, obtain the ideal polishing trajectory.

[0017] Further, step S11 includes:

[0018] Obtain the surface equation f1(x, y) of the main mirror body;

[0019] Obtain the surface equation f3(x, y) of the secondary mirror;

[0020] Obtain the central hole equation f h (x, y) in the multi-faceted coaxial element;

[0021] Based on the surface equation f1(x, y) of the primary mirror, the surface equation f3(x, y) of the secondary mirror, and the central hole equation f h (x, y), determine the coordinate vectors of each point in the multi-faceted coaxial element , where the sagittal height z is:

[0022] ;

[0023] Back up the sagittal height z three times to obtain the first sagittal height z, the second sagittal height z, and the third sagittal height z.

[0024] Furthermore, in step S12, based on the first sagittal height z, generate the first trajectory segment through the following formula:

[0025] ;

[0026] where z1 represents the first sagittal height z, R3 represents the semi-aperture of the secondary mirror, None represents a null value, Pathize represents a function that makes a point set into a raster trajectory, and (ε1, η1, γ1) represents the coordinate values in the first trajectory segment.

[0027] Furthermore, in step S13, based on the second sagittal height z, generate the second trajectory segment through the following formula:

[0028] ;

[0029] where z2 represents the second sagittal height z, and (ε2, η2, γ2) represents the coordinate values in the second trajectory segment.

[0030] Furthermore, in step S14, based on the third sagittal height z, generate the third trajectory segment through the following formula:

[0031] ;

[0032] where z3 represents the third sagittal height z, (ε3, η3, γ3) represents the coordinate values in the third trajectory segment, and R h represents the semi-aperture of the central hole.

[0033] Furthermore, in step S15, splice the first trajectory segment, the second trajectory segment, and the third trajectory segment through the following formula:

[0034] ;

[0035] Among them, (ε, η, γ) represents the coordinates of each trajectory point in the ideal polishing trajectory, and cat represents the function for linking parameter vectors.

[0036] Furthermore, the process of determining the normal vectors of each point on the mirror body in the multi-faceted coaxial element in step S2 includes: establishing a global machining coordinate system when machining the multi-faceted coaxial element, with the intersection point of the slave mirror surface of the multi-faceted coaxial element and the reference horizontal plane as the origin of the global machining coordinate system; the common axis of rotation of the multi-faceted coaxial element is the Z-axis of the global machining coordinate system, and the direction with a positive cosine of the angle between the normal vectors of each point on the multi-faceted coaxial mirror surface and the Z-axis is the positive direction of the global Z-axis, and the X-axis and Y-axis of the machining coordinate system of the machining tool itself are the X-axis and Y-axis of the global machining coordinate system; calculating the normal vectors of each point on the mirror body in the multi-faceted coaxial element in the global machining coordinate system.

[0037] Furthermore, the machining coordinate system is translated in the Z-axis direction relative to the global machining coordinate system , represents the normal vectors of each point on the mirror body in the multi-faceted coaxial element, and Δ represents the polishing gap.

[0038] Furthermore, in step S3, the coordinates of each trajectory point in the actual polishing trajectory are determined by the following formula:

[0039] ;

[0040] Among them, represents the coordinates of each trajectory point in the actual polishing trajectory, represents the coordinates of each trajectory point in the ideal polishing trajectory, ;

[0041] The machining attitude is determined by the following formula:

[0042] ;

[0043] Among them, represents the machining attitude.

[0044] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0045] The pose calculation method for integrated machining of magnetorheological polishing multi-faceted coaxial elements in the present invention for creating an invention is for the same process section of multi-faceted coaxial zoning scanning machining, that is, the main mirror is machined in a single scanning trajectory, effectively avoiding the problems of pose conversion at the seam and the excessive requirements for equipment and algorithms. In addition, the present invention adopts a circular arm (i.e., the main mirror and the secondary mirror form a ring in the top view, and the circular arm is a part of this ring) circumferential trajectory strategy, avoiding redundant machining time, not requiring consideration of the requirements for high consistency of the trajectory due to the concentric and coaxial tolerances of multi-faceted coaxial elements, and not having excessive requirements for algorithms and equipment, thus shortening the manufacturing cycle for the polishing of large-aperture multi-faceted coaxial elements. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] The drawings forming a part of the present invention for creating an invention are used to provide a further understanding of the present invention for creating an invention. The schematic embodiments and descriptions thereof of the present invention for creating an invention are used to explain the present invention for creating an invention and do not constitute an improper limitation to the present invention for creating an invention. In the drawings:

[0047] Figure 1 is a schematic flow chart of the pose calculation method for integrated machining of magnetorheological polishing multi-faceted coaxial elements according to an embodiment of the present invention for creating an invention;

[0048] Figure 2 is a schematic diagram of the first trajectory segment according to an embodiment of the present invention for creating an invention;

[0049] Figure 3 is a schematic diagram of the second trajectory segment according to an embodiment of the present invention for creating an invention;

[0050] Figure 4 is a schematic diagram of the third trajectory segment according to an embodiment of the present invention for creating an invention;

[0051] Figure 5 is a schematic diagram of the polishing trajectory according to an embodiment of the present invention for creating an invention;

[0052] Figure 6 is a coordinate system relationship diagram according to an embodiment of the present invention for creating an invention;

[0053] Figure 7 is a tool pose vector diagram according to an embodiment of the present invention for creating an invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0054] In order to make the objectives, technical solutions and advantages of the present invention for creating an invention clearer and more understandable, the following further details the present invention for creating an invention in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention for creating an invention and do not constitute a limitation to the present invention for creating an invention.

[0055] It should be noted that, without conflict, the embodiments and features in the embodiments of the present invention for creating an invention can be combined with each other.

[0056] In the description of the present invention, it should be understood that the orientation or positional relationship indicated by the terms "center", "longitudinal", "transverse", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings. It is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as a limitation to the present invention. In addition, the terms "first", "second", etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly specifying the quantity of the indicated technical features. Thus, the features defined with "first", "second", etc. may explicitly or implicitly include one or more of such features. In the description of the present invention, unless otherwise stated, the meaning of "a plurality" is two or more.

[0057] In the description of the present invention, it should be noted that unless otherwise clearly specified and limited, the terms "installed", "connected", "connected" should be understood in a broad sense. For example, it may be a fixed connection, a detachable connection, or an integral connection; it may be a mechanical connection or an electrical connection; it may be directly connected, or indirectly connected through an intermediate medium, and it may be the communication inside two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood through specific situations.

[0058] The present invention will be described in detail below with reference to the drawings and in conjunction with embodiments.

[0059] As Figure 1 shown, the posture calculation method for integral machining of a magnetorheological polishing multi-faceted co-body element according to an embodiment of the present invention includes:

[0060] S1: Determine the trajectory segments corresponding to each mirror body during machining according to the surfaces of the mirror bodies in the multi-faceted co-body element to be machined; combine the trajectory segments corresponding to each mirror body to obtain the ideal polishing trajectory during machining.

[0061] In some embodiments, step S1 includes:

[0062] S11: Obtain the surface equations of the main mirror and the slave mirror in the multi-faceted co-body element. Specifically, step S11 includes:

[0063] Obtain the surface equation f1(x, y) of the main mirror;

[0064] Obtain the surface equation f3(x, y) of the slave mirror;

[0065] Obtain the central hole equation f of the multi-faceted co-body elementh (x, y);

[0066] Based on the main mirror surface equation f1(x, y), the slave mirror surface equation f3(x, y), and the central hole equation f h (x, y), determine the coordinate vectors of each point in the multi-faceted co-body element , where the sagittal height z is:

[0067] ;

[0068] Back up the sagittal height z three times to obtain the first sagittal height z, the second sagittal height z, and the third sagittal height z.

[0069] S12: Taking the single-sided one-ring arm of the main mirror as a reference, generate the first trajectory segment. Specifically, based on the first sagittal height z, generate the first trajectory segment through the following formula:

[0070] ;

[0071] where z1 represents the first sagittal height z, R3 represents the semi-aperture of the slave mirror, None represents a null value, Pathize represents a function that makes a point set into a grating trajectory, and (ε1, η1, γ1) represents the coordinate values in the first trajectory segment.

[0072] In a certain embodiment, the architecture of the function that makes a point set into a grating trajectory is as follows:

[0073] Set three empty vectors a, b, and c, scan the rows and columns of the sagittal height z matrix. If i in the i-th row is odd, scan forward and sequentially append the non-empty data of (x, y, z) in this row to the end of the a, b, and c vectors. If i in the i-th row is even, scan backward and sequentially append the non-empty data of (x, y, z) in this row to the end of the a, b, and c vectors.

[0074] S13: Taking the remaining ring arms of the main mirror except for the ring arm in S12 as a reference, generate the second trajectory segment. Specifically, based on the second sagittal height z, generate the second trajectory segment through the following formula:

[0075] ;

[0076] where z2 represents the second sagittal height z, and (ε2, η2, γ2) represents the coordinate values in the second trajectory segment.

[0077] S14: Taking the slave mirror as a reference, generate the third trajectory segment. Specifically, based on the third sagittal height z, generate the third trajectory segment through the following formula:

[0078] ;

[0079] Among them, z3 represents the third sagitta z, (ε3, η3, γ3) represents the coordinate values in the third trajectory segment, and R h represents the semi-aperture of the central hole.

[0080] S15: Obtain the ideal polishing trajectory based on the first trajectory segment, the second trajectory segment, and the third trajectory segment obtained in steps S12 to S14. Specifically, splice the first trajectory segment, the second trajectory segment, and the third trajectory segment through the following formula:

[0081] ;

[0082] Among them, (ε, η, γ) represents the coordinates of each trajectory point in the ideal polishing trajectory, and cat represents the function for concatenating parameter vectors.

[0083] S2: Set the polishing gap during machining of the machining tool, and determine the normal vectors of each point on the mirror body in the multi-faceted co-body element.

[0084] In some embodiments, the process of determining the normal vectors of each point on the mirror body in the multi-faceted co-body element includes: establishing a global machining coordinate system for machining the multi-faceted co-body element,

[0085] Among them, the intersection point of the body mirror surface and the reference horizontal plane in the multi-faceted co-body element is the origin of the global machining coordinate system; the common rotation axis of the multi-faceted co-body element is the Z-axis of the global machining coordinate system, the direction with the cosine of the angle between the normal vectors of each point on the multi-faceted co-body mirror surface being positive is the positive direction of the global Z-axis, and the X-axis and Y-axis of the machining coordinate system of the machining tool itself are the X-axis and Y-axis of the global machining coordinate system. The machining coordinate system is translated in the Z-axis direction relative to the global machining coordinate system by , represents the normal vectors of each point on the mirror body in the multi-faceted co-body element, and Δ represents the polishing gap.

[0086] Calculate the normal vectors of each point on the mirror body in the multi-faceted co-body element in the global machining coordinate system.

[0087] S3: Obtain the actual polishing trajectory of the machining tool and the machining posture of the machining tool based on the ideal polishing trajectory obtained in step S1, and the polishing gap and the normal vectors of each point obtained in step S2. Specifically, determine the machining position through the following formula:

[0088] ;

[0089] Among them, represents the coordinates of each trajectory point in the actual polishing trajectory, represents the coordinates of each trajectory point in the ideal polishing trajectory, ;

[0090] Determine the machining posture through the following formula:

[0091] ;

[0092] Among them, represents the machining posture.

[0093] In one embodiment, it is necessary to set the initial pose of the machining tool, and the initial pose includes the initial machining position , and the initial machining posture . During the actual machining process, first control the machining tool to adjust to the initial pose. After the initial pose is set, new machining positions and postures are obtained according to the formulas for determining the machining position and posture for machining.

[0094] S4: Combine the actual polishing trajectory and the machining posture obtained in step S3 to obtain the machining pose of the machining tool.

[0095] During the optical machining process, the machining pose of the machining tool includes the machining position of the machining tool and the machining posture of the machining tool. The machining position of the machining tool is the coordinate of each trajectory point in the actual polishing trajectory of the machining tool. Therefore, when the actual polishing trajectory and the machining posture of the machining tool are determined, the machining pose of the machining tool during the machining process can be directly obtained.

[0096] To clearly illustrate the method for calculating the integrated machining pose of a magnetorheological polishing multi-faceted coaxial element provided by the present invention, an embodiment is provided.

[0097] Embodiment: The method for calculating the integrated machining pose of a magnetorheological polishing multi-faceted coaxial element described in this embodiment includes:

[0098] S1: Determine the trajectory segments corresponding to each mirror body during machining according to the surfaces of each mirror body in the multi-faceted coaxial element to be machined; combine the trajectory segments corresponding to each mirror body to obtain the ideal polishing trajectory during machining.

[0099] In this embodiment, step S1 includes:

[0100] S11: Obtain the surface equations of the main mirror and the slave mirror in the multi-faceted coaxial element. Specifically, step S11 includes:

[0101] Obtain the surface equation f1(x, y) of the main mirror as:

[0102] ;

[0103] Obtain the surface equation f3(x, y) of the slave mirror as:

[0104] ;

[0105] Obtain the central hole equation f h (x, y) of the multi-faceted coaxial element as:

[0106] ;

[0107] Based on the surface equation f1(x, y) of the main mirror, the surface equation f3(x, y) of the slave mirror, and the central hole equation f h (x, y), determine the coordinate vectors of each point in the multi-faceted co-body element , where the sagittal height z is:

[0108] ;

[0109] Back up the sagittal height z three times to obtain the first sagittal height z, the second sagittal height z, and the third sagittal height z.

[0110] S12: Taking one ring arm on one side of the main mirror as a reference, generate the first trajectory segment. Specifically, based on the first sagittal height z, generate the first trajectory segment as shown in Figure 2 as follows:

[0111] ;

[0112] where the semi-aperture R3 of the slave mirror = 1000, and the pseudo-code of the function Pathize for the point set to be the grating trajectory can be:

[0113]

[0114] where z.rows is the number of rows of the sagittal height z matrix, and z.cols is the number of columns of the sagittal height z matrix.

[0115] S13: Taking the remaining ring arms of the main mirror except the ring arm in S12 as a reference, generate the second trajectory segment. Specifically, based on the second sagittal height z, generate the second trajectory segment as shown in Figure 3 as follows:

[0116] .

[0117] S14: Taking the slave mirror as a reference, generate the third trajectory segment. Specifically, based on the third sagittal height z, generate the third trajectory segment as shown in Figure 4 as follows:

[0118] ;

[0119] where the semi-aperture R h = 500.

[0120] S15: According to the first trajectory segment, the second trajectory segment, and the third trajectory segment obtained in steps S12 to S14, obtain as shown in Figure 5The ideal polishing trajectory shown. Specifically, the first trajectory segment, the second trajectory segment, and the third trajectory segment are spliced by the following formula:

[0121] ;

[0122] If the main mirror and the slave mirror are processed in two separate work sections using the traditional method, according to the previous overall processing strategy, there will be redundant time:

[0123] ;

[0124] where, T m represents the minimum dwell time for each dwell point due to the machine tool motion performance limitation, P represents the number of idle running dwell points, and as the semi-aperture R3 = 1000 of the slave body mirror increases, the redundant processing time increases in a quadratic trend. That is, when the semi-aperture R3 of the slave body mirror is 2000, the redundant time T ry ≈8.72 h.

[0125] S2: Set the polishing gap during machining of the machining tool, and determine the normal vectors of each point on the mirror body in the multi-faceted co-body element.

[0126] In this embodiment, the polishing gap Δ = 2 is set. Taking the intersection point of the coaxial main mirror and the curved surface of the secondary mirror in the multi-faceted co-body element as the tool origin, and taking the coaxial axis of the co-body mirror as the global Z-axis, and the direction with the cosine of the angle between the normal vectors of each point on the co-body mirror surface being positive as the positive direction of the global Z-axis, combined with the X-axis and Y-axis of the machining coordinate system of the machining tool itself, determine the global machining coordinate system as shown in Figure 6 ; Figure 6 where S w represents the global machining coordinate system, S T represents the machining coordinate system, and the normal vectors of each point on the mirror body in the multi-faceted co-body element in the global machining coordinate system are obtained by the following formula:

[0127] ;

[0128] where, represents the unnormalized surface normal vector, F1 = f1(x, y) - z, F3 = f3(x, y) - z, F 1x , F 1y , F 1z , are the partial derivatives of the equation F1 with respect to x, y, z, F 3x , F 3y , F 3z , are the partial derivatives of the equation F3 with respect to x, y, z, and the equation F is defined as:

[0129] .

[0130] Among them, the Z-axis direction of the machining coordinate system is translated relative to the global machining coordinate system .

[0131] S3: According to the ideal polishing trajectory obtained in step S1, the polishing gap obtained in step S2, and the normal vectors of each point, obtain the actual polishing trajectory of the machining tool and the machining posture of the machining tool.

[0132] In this embodiment, first set the initial pose of the machining tool, and the initial pose includes the initial machining position , and the initial machining posture .

[0133] Determine the machining position through the following formula :

[0134] ;

[0135] Among them, F x , F y and F z are the partial derivatives of the equation F with respect to x, y, and z respectively;

[0136] Determine the machining posture through the following formula :

[0137] .

[0138] S4: Combine the actual polishing trajectory and the machining posture obtained in step S3 to obtain the machining pose of the machining tool. The discretized pose obtained in this embodiment is as Figure 7 shown.

[0139] It should be understood that various forms of the flow shown above can be used, with steps reordered, added, or deleted. For example, the steps described in the disclosure of the present invention can be executed in parallel, sequentially, or in a different order, as long as the desired results of the technical solution disclosed in the present invention can be achieved, and no limitation is imposed herein.

[0140] The above specific embodiments do not constitute a limitation to the protection scope of the present invention. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A posture calculation method for integrated machining of a magnetorheological polishing multi-faceted co-body element, characterized in that, Including: S1: Determine the trajectory segments corresponding to each mirror body during processing according to the surfaces of the mirror bodies in the multi-faceted coaxial body element to be processed; Combine the trajectory segments corresponding to each mirror body to obtain the ideal polishing trajectory during processing; Step S1 includes: S11: Obtain the surface equations of the main mirror and the slave mirror in the multi-faceted coaxial body element; S12: Generate a first trajectory segment with a single ring arm on one side of the main mirror as a reference; S13: Generate a second trajectory segment with the remaining ring arms of the main mirror except the ring arm in S12 as a reference; S14: Generate a third trajectory segment with the slave mirror as a reference; S15: Obtain the ideal polishing trajectory according to the first trajectory segment, the second trajectory segment, and the third trajectory segment obtained in steps S12 to S14; S2: Set the polishing gap during processing of the processing tool, and determine the normal vectors of each point of the mirror body in the multi-faceted coaxial body element; S3: Obtain the actual polishing trajectory of the processing tool and the processing posture of the processing tool according to the ideal polishing trajectory obtained in step S1, and the polishing gap and the normal vectors of each point obtained in step S2; S4: Combine the actual polishing trajectory and the processing posture obtained in step S3 to obtain the processing position and posture of the processing tool.

2. The pose calculation method for integrated machining of a magnetorheological polishing multi-faceted coaxial element according to claim 1, characterized in that Step S11 includes: Obtain the surface equation f1(x, y) of the main mirror; Obtain the surface equation f3(x, y) of the slave mirror; Obtain the central hole equation f in the multi-faceted coaxial element h (x, y); Based on the surface equation f1(x, y) of the main body mirror, the surface equation f3(x, y) of the slave body mirror, and the central hole equation f h (x, y), determine the coordinate vectors of each point in the multi-faceted co-body element , where the sagittal height z is: ； Back up the sagittal height z three times to obtain the first sagittal height z, the second sagittal height z, and the third sagittal height z.

3. The pose calculation method for integrated machining of a magnetorheological polishing multi-faceted common body element according to claim 2, wherein, In step S12, based on the first sagittal height z, generate the first trajectory segment through the following formula: ； where z1 represents the first sagittal height z, R3 represents the semi-aperture of the slave mirror, None represents a null value, Pathize represents a function to make a point set into a raster trajectory, and (ε1, η1, γ1) represents the coordinate values in the first trajectory segment.

4. The pose calculation method for integrated machining of a magnetorheological polishing multi-faceted coaxial element according to claim 3, characterized in that, In step S13, based on the second sagittal height z, generate the second trajectory segment through the following formula: ； where z2 represents the second sagittal height z, and (ε2, η2, γ2) represents the coordinate values in the second trajectory segment.

5. The pose calculation method for integral machining of a magnetorheological polishing multi-faceted coaxial component according to claim 4, characterized in that In step S14, based on the third sagittal height z, generate the third trajectory segment through the following formula: ； where z3 represents the third sagitta z, (ε3, η3, γ3) represents the coordinate values in the third trajectory segment, and R h represents the semi-aperture of the central hole.

6. The pose calculation method for integral machining of a magnetorheological polishing multi-faceted coaxial element according to claim 5, wherein In step S15, splice the first trajectory segment, the second trajectory segment, and the third trajectory segment through the following formula: ； where (ε, η, γ) represents the coordinates of each trajectory point in the ideal polishing trajectory, and cat represents a function to link parameter vectors.

7. The pose calculation method for integrated machining of a magnetorheological polishing multi-faceted coaxial element according to claim 6, wherein The process of determining the normal vectors of each point of the mirror body in the multi-faceted coaxial body element in step S2 includes: Establish a global processing coordinate system for processing the multi-faceted coaxial body element. The intersection point of the slave mirror surface of the multi-faceted coaxial body element and the reference horizontal plane is the origin of the global processing coordinate system; the common rotation axis of the multi-faceted coaxial body element is the Z axis of the global processing coordinate system, and the direction with a positive cosine of the angle between the normal vectors of each point on the surface of the multi-faceted coaxial body element and the Z axis is the positive direction of the global Z axis. The X axis and Y axis of the processing coordinate system of the processing tool itself are the X axis and Y axis of the global processing coordinate system; Calculate the normal vectors of each point on the mirror body in the multi-faceted coaxial element in the global machining coordinate system.

8. The pose calculation method for integral machining of a magnetorheological polishing multi-faceted coaxial element according to claim 7, characterized in that, The Z-axis direction of the machining coordinate system is translated relative to the Z-axis direction of the global machining coordinate system , represents the normal vectors of the points on the mirror body in the multi-faceted common body element, and Δ represents the polishing gap.

9. The pose calculation method for integral machining of a magnetorheological polishing multi-faceted coaxial element according to claim 8, characterized in that In step S3, determine the coordinates of each trajectory point in the actual polishing trajectory by the following formula: ； Among them, represents the coordinates of each trajectory point in the actual polishing trajectory, represents the coordinates of each trajectory point in the ideal polishing trajectory, ; Determine the machining posture by the following formula: ； Among them, represents the machining posture.

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