A linear elastic mechanism posture transfer modeling method for a dual-degree-of-freedom dynamic derivative test device

Abstract

The present invention belongs to the field of mechanism dynamics modeling, and discloses a linear elastic mechanism posture transfer modeling method of a dual-degree-of-freedom dynamic derivative test device. The method is based on an existing dual-degree-of-freedom dynamic derivative test device mechanism, analyzes the mechanism motion chain; solves the rigid body posture transfer matrix of the mechanism according to the motion chain, and obtains the input-output relationship of the mechanism posture; corrects the elastic contraction and bending of the component after being stressed by establishing an elastic unit on the kinematic pair; corrects the elastic amount of the component to the posture transfer model of the mechanism, and obtains the mechanism posture transfer matrix taking into account the elastic deformation; thereby obtaining the linear elastic mechanism posture transfer model of the dual-degree-of-freedom dynamic derivative test device under complex aerodynamic loads; the invention has good description performance for the elastic deformation of the device mechanism caused by time-varying aerodynamic loads, the calculation formula is concise and reliable, and the solution is simple, which not only clarifies the posture transfer characteristics of the linear elastic mechanism, but also meets the timeliness requirements of motion feedback control.

Description

Technical Field

[0001] The invention belongs to the field of mechanism dynamics modeling, and relates to a linear elastic mechanism posture transfer modeling method of a dual-degree-of-freedom dynamic derivative test device. Background Art

[0002] In the current era, ultra-high-speed performance and high-maneuverability flight capabilities have become the strategic direction of high-end aircraft. In order to ensure that the aircraft has sufficient dynamic stability under such conditions, its flight mission places higher requirements on the aerodynamic performance of the aircraft. Therefore, the evaluation of the dynamic stability of the aircraft has become an indispensable part of the model development process. Through the wind tunnel dynamic derivative test, the aircraft model is placed under simulated conditions, and the dynamic derivative of the aircraft model is accurately measured to form a reliable criterion for the dynamic stability of the aircraft. Therefore, the calculation ability and accuracy of the dynamic derivative test have become the decisive factors in the dynamic stability evaluation, and play an important role in the development of high-end aircraft models.

[0003] Existing dynamic derivative tests usually fix the aircraft model to the end of the mechanical mechanism, control the motion posture of the aircraft model through the mechanical mechanism, synchronously measure the multi-dimensional aerodynamic force and multi-degree-of-freedom angular displacement of the aircraft model, and obtain the required dynamic derivatives through signal processing and solution. However, when measuring the stability boundary of the aircraft, such as under conditions such as large angle of attack and transonic nonlinear stage, the mechanical mechanism will inevitably be impacted by the complex disturbance flow field and deformed, resulting in errors and lags in the force and displacement output, unclear posture transfer characteristics, and the aircraft model movement deviates from the target value, thereby affecting the accuracy of dynamic stability assessment. By modeling the posture transfer of the linear elastic mechanism, the transmission performance and dynamic characteristics of the mechanical structure can be more accurately described, so that the impact of the complex disturbance flow field on it can be considered in the design stage. At the same time, it is also conducive to improving the control accuracy of the aircraft model movement, thereby ensuring the accuracy of dynamic derivative measurement.

[0004] In the paper "Kinematic Calibration and Elastic Dynamics Analysis of Parallel Angular Tracking Mechanism" published in 2020, Song Yimin's team from Tianjin University proposed a multi-flexible body dynamics modeling method. The shape function of the three-dimensional isoparametric beam unit taking into account the curvature was used to derive the dynamic equation of the spatial beam unit based on the Lagrange equation, and the multi-flexible body dynamics model of the parallel angular tracking mechanism was obtained, thereby realizing the analysis and calibration of the posture transfer characteristics of the parallel angular tracking mechanism. However, although the established model is relatively accurate, the calculation is relatively complex, and it is difficult to achieve online tracking of the linear elasticity of the mechanism. In the paper "Elastic Dynamics Modeling and Analysis of 3-PRS Parallel Power Head" published in 2017 by Zhang Jun's team from Anhui University of Technology, a system elastic dynamics model of the 3-PRS parallel power head was established based on the principle of deformation superposition and the idea of ​​dynamic substructure integration. By taking into account the elastic deformation of each kinematic hinge and the structural flexibility of the branch body, the deformation coordination relationship was used to establish the system elastic dynamics model of the 3-PRS parallel power head, which can realize the calculation of the posture transfer and dynamic response of the system. However, the established model still needs to be decomposed by the finite element method, and the accuracy is related to the number of decomposed units, which makes the high-precision solution more complicated.

[0005] In summary, the current system elastic modeling method can realize the accurate calculation of posture transfer taking into account the system deformation. However, its calculation is generally complicated and difficult to be directly used for posture transfer solution and motion feedback control in dynamic derivative test mechanisms. Therefore, there is an urgent need for a posture transfer modeling method for linear elastic mechanisms of dual-degree-of-freedom dynamic derivative test devices that can be quickly solved. Summary of the invention

[0006] The main technical problem solved by the present invention is to overcome the shortcomings of the prior art, aiming at the problem that the elastic deformation of the dynamic derivative test device under complex aerodynamic loads causes errors and lags in force and displacement output, and the existing elastic modeling method is complex and inefficient in calculating the posture transfer relationship, and does not meet the requirements of posture transfer solution and motion feedback control of the dynamic derivative experimental mechanism. A posture transfer modeling method for a linear elastic mechanism of a dual-degree-of-freedom dynamic derivative test device is invented. The invention is based on the existing dual-degree-of-freedom dynamic derivative test device mechanism, and analyzes the mechanism motion chain; solves the rigid body posture transfer matrix of the mechanism according to the motion chain to obtain the input-output relationship of the mechanism posture; corrects the elastic contraction and bending of the component after being subjected to force by establishing an elastic unit on the kinematic pair; corrects the elastic amount of the component to the posture transfer model model of the mechanism to obtain the mechanism posture transfer matrix taking into account the elastic deformation; thereby, the linear elastic mechanism posture transfer model of the dual-degree-of-freedom dynamic derivative test device under complex aerodynamic loads is obtained; the invention has good descriptive performance for the elastic deformation of the device mechanism caused by time-varying aerodynamic loads, the calculation formula is concise and reliable, and the solution is simple. It not only clarifies the posture transfer characteristics of the linear elastic mechanism, but also meets the timeliness requirements of motion feedback control.

[0007] The technical solution of the present invention:

[0008] A linear elastic mechanism posture transfer modeling method for a dual-degree-of-freedom dynamic derivative test device is disclosed. Firstly, based on the existing dual-degree-of-freedom dynamic derivative test device mechanism, the motion chain of the device mechanism is analyzed; secondly, the motion chain analysis is performed through the vector closed-loop method to calculate and solve the posture transfer relationship of the device mechanism; then, elastic correction is performed on the key components of the device mechanism; and the posture transfer model of the dual-degree-of-freedom dynamic derivative test device after solution and correction is obtained.

[0009] The specific steps are as follows:

[0010] Step 1: Based on the dual-degree-of-freedom dynamic derivative test device, describe and analyze the kinematic chain;

[0011] The dual-degree-of-freedom dynamic derivative test device comprises components such as a piezoelectric driver, a transmission connecting rod, a moving bump, a rotating end and a rotating base. There are four motion branches in total, which are connected in parallel to the rotating end. The structures of the motion branches are the same. A cross axis is connected between the rotating end and the rotating base, and there are rotational degrees of freedom in two orthogonal directions of yaw and roll. Taking one of the motion branches as an example, there is a piezoelectric driver as a linear displacement output. The piezoelectric driver and the moving protrusion are matched through a spherical surface. The moving protrusion can be offset along the spherical surface while transmitting the displacement output of the piezoelectric driver, thereby providing a rotational degree of freedom. The moving protrusion fits with the transmission connecting rod through the cylindrical surface to transmit the linear displacement of the driver to one end of the transmission connecting rod. The transmission connecting rod is connected to the rotating base through the cylindrical shaft hole as a fulcrum of movement. The other end of the transmission connecting rod is a matching plane, which is in point-to-point contact with the rolling ball on the rotating end. The transmission connecting rod and the moving protrusion together amplify the linear displacement of the driver at one end and convert it into a swing arc at the other end. The four motion branches are annularly symmetrically arrayed around the rotating end, and the matching plane of the transmission connecting rod is in contact with the four rolling balls on the rotating end.

[0012] The calculation formula of the motion freedom of the dual-degree-of-freedom dynamic derivative test device is as follows:

[0013] (1)

[0014] Among them, DOF represents the degree of freedom of the dynamic derivative test device, n represents the number of components in the device kinematic chain, represents the low number of sub-constraints, represents the number of high pair constraints, represents the driving degree of freedom;

[0015] Step 2: Solve the posture transfer relationship of the dual-degree-of-freedom dynamic derivative test device;

[0016] According to the calculation results of degrees of freedom, two of the motion branches of the dual-degree-of-freedom dynamic derivative test device are active branches. The piezoelectric drivers of the two branches are AB and A'B', the moving bumps are BC and B'C', the transmission links are CDE and C'D'E', CDE includes CD segment and DE segment, C'D'E' is similar, the rotation end is EE'FG, including EF segment, E'F segment and FG segment, and EF segment and E'F segment are perpendicular to FG segment; take point G as the coordinate origin, establish the rectangular coordinate system Gxyz, where Gx is the roll axis, Gy is the pitch axis, Gz is the is the yaw axis, the FG segment at the end of the rotation rotates around Gz, and coincides with the roll axis Gx in the initial state, the EF segment and the E'F segment rotate around the FG segment, so the end points E and E' can rotate around the roll axis Gx and the yaw axis Gz relative to the end point G; D and D' are two fixed points in the rectangular coordinate system Gxyz, CDE and C'D'E' are hinged to the rotating base at D and D' respectively, and are hinged to BC and B'C' at C and C' respectively; BC and B'C' are hinged to AB and A'B' at B and B' respectively, and A and A' are two fixed points in the rectangular coordinate system Gxyz;

[0017] According to the closed-loop vector method, the vector closed loop of the kinematic chain is obtained, and its vector closed loop equation is as follows:

[0018] (2)

[0019] in, , are the coordinates of point E and point E', , , , are the coordinates of point A, point A', point G and point F, , , , , , , , , , The vector lengths are AB, BC, CD, DE, A'B', B'C', C'D', D'E', EF, and GF respectively;

[0020] Select the kinematic chain AB-BC-CD-DE-EF-FG as the analysis object, decompose the vector orthogonally, and expand it to get the following formula:

[0021] (3)

[0022] in, , , , , Respectively represent the vector lengths of AB, BC, CD, DE, and EF, is the projection of GE on plane oxy, , , , Respectively represent the angles between BC, CD, DE, GE and the roll axis, , is the component of AD along the roll axis and pitch axis, , is the component of GD along the roll axis and pitch axis, is the initial angle between DE and the roll axis, for Component along the pitch axis;

[0023] Through the vector closed-loop equation, the displacement input-output relationship under a single branch chain can be obtained. Select the A'B'-B'C-C'D'-D'E'-EF-FG branch chain and establish a vector closed-loop equation of the same form. When the output of the piezoelectric driver is known, the angle and displacement of the internal components of the branch chain can be calculated;

[0024] Arranging equation (2) and equation (3) yields the following equation:

[0025] (4)

[0026] in, , , , , Respectively represent the vector lengths of A'B', B'C', C'D', D'E', and E'F', is the projection of GE' on plane oxy, , , , They represent the angles between B'C', C'D', D'E', GE' and the roll axis, , is the component of A'D' along the roll axis and pitch axis, , is the component of GD' along the roll axis and pitch axis, is the initial angle between D'E' and the roll axis, for Component along the pitch axis;

[0027] Select the kinematic chain EF-EE'FG-E'F and construct the displacement transfer equation as follows:

[0028] (5)

[0029] in, , They are , The component along the pitch axis, , , , are the vector lengths of EF, E'F, EG, and E'G respectively, , are the angles between EF, E'F and the pitch axis, , are the angles between EG and E'G and the roll axis, respectively, which can be solved by equation (3). Based on the geometric relationship, the roll angle and yaw angle are calculated as follows:

[0030] (6)

[0031] In the formula, , are the roll angle and yaw angle respectively, is the initial angle between EF and the pitch axis, is the initial angle between EG and the roll axis;

[0032] Step 3: Construct the elastic deformation correction method of the mechanism components of the dual-degree-of-freedom dynamic derivative test device;

[0033] When the mechanism is subjected to load during transmission, elastic deformation will occur, and the deformation of its components will be transmitted to the next kinematic pair through the kinematic chain, thereby affecting the posture mapping relationship between the driver and the rotation end; In order to simplify the calculation process of component deformation transmission in the dual-degree-of-freedom dynamic derivative test device mechanism, a flexible hinge model is proposed for the compressive elastic deformation of the components. Two groups of spring units with different stiffness are set in the orthogonal direction on the rotation pair at the hinge between the components, which are used for the compressive elastic deformation of the components at both ends of the equivalent kinematic pair; its formula is expressed as follows:

[0034] [ δ x δ y ] = [ F x / k x F y / k y ] (7)

[0035] in, , is the deformation along two orthogonal directions, , is the load along two orthogonal directions, , is the stiffness along two orthogonal directions; , is the deformation amount and deformation rate along the normal vector of the contact surface;

[0036] Aiming at the bending deformation and mechanical damping of components, a flexible damping contact model is proposed. A spring damping unit is set along the normal vector on the contact surface to be used for the bending elastic deformation and mechanical damping of the equivalent transmission connecting rod. The formula is described as follows:

[0037] (8)

[0038] in, represents the load along the contact surface normal vector, , is the stiffness and damping along the normal vector of the contact surface;

[0039] Step 4: Solve the elastically corrected dual-degree-of-freedom dynamic derivative test device mechanism posture transfer model;

[0040] Since the elastic deformation of the components of the dual-degree-of-freedom dynamic derivative test device is equivalent to the deformation of the flexible hinge model and the flexible damping contact model on the revolute pair and the point-to-surface contact pair, in the process of calculating the posture transfer, it is only necessary to supplement the deformation of the revolute pair and the point-to-surface contact pair on the basis of formula (3) based on the geometric relationship to obtain the elastically corrected posture transfer relationship of the device mechanism without considering the specific deformation of the components; the corrected posture transfer model is as follows:

[0041] (9)

[0042] in, , is the deformation of the revolute pair C and D along the roll axis direction, , is the deformation of the revolute joint C and D along the pitch axis, It is the deformation of point-to-surface contact pair E along the normal direction of the contact surface.

[0043] The beneficial effect of the present invention is to provide a linear elastic mechanism posture transfer modeling method for a dual-degree-of-freedom dynamic derivative test device, which can describe the linear elastic deformation of the dual-degree-of-freedom dynamic derivative test device mechanism and its posture transfer relationship under aerodynamic load, and can be used in the motion feedback control of the dual-degree-of-freedom dynamic derivative test device mechanism. By performing a motion chain analysis on the dual-degree-of-freedom dynamic derivative test device mechanism; secondly solving the posture transfer relationship of the dynamic derivative test device mechanism; then constructing an elastic deformation correction method for the dual-degree-of-freedom dynamic derivative test device mechanism components; and then solving the posture transfer model of the dual-degree-of-freedom dynamic derivative test device mechanism after elastic correction; the invention can accurately solve the influence of aerodynamic loads on the posture transfer of the dynamic derivative test device mechanism under the high-speed flow field of the aircraft model, and ensure the accuracy of attitude control, ensure the accuracy of dynamic derivative measurement, and greatly improve the efficiency of dynamic derivative testing. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a method flow chart of a method for modeling a linear elastic mechanism posture transfer of a dual-degree-of-freedom dynamic derivative test device of the present invention;

[0045] Figure 2 It is a structural schematic diagram of the dual-degree-of-freedom dynamic derivative test device of the present invention;

[0046] Figure 3 It is a cross-sectional schematic diagram of a dual-degree-of-freedom dynamic derivative test device of the present invention;

[0047] Figure 4 It is a cross-sectional schematic diagram of a dual-degree-of-freedom dynamic derivative test device of the present invention;

[0048] Figure 5 Schematic diagram of the flexible hinge model of the dual-degree-of-freedom dynamic derivative test device of the present invention

[0049] Figure 6 It is a schematic diagram of the flexible damping contact model of the dual-degree-of-freedom dynamic derivative test device of the present invention.

[0050] Figure 7 This is the solution result of the posture transfer of the dual-degree-of-freedom dynamic derivative test device of the present invention. DETAILED DESCRIPTION

[0051] The specific implementation of the present invention is described in detail below in combination with the technical scheme and the accompanying drawings.

[0052] The embodiment takes a dual-degree-of-freedom forced vibration dynamic derivative test system designed for a 2.4m wind tunnel as an example, and is described as follows in conjunction with the accompanying drawings:

[0053] Step 1: Perform kinematic chain analysis on the dual-degree-of-freedom dynamic derivative test device;

[0054] The dual-degree-of-freedom dynamic derivative test device includes components such as a piezoelectric driver, a transmission connecting rod, a moving bump, a rotating end, and a rotating base. There are four motion branches in total, which are connected in parallel to the rotating end. The rotating end is connected to the rotating base through a cross axis to form rotational degrees of freedom in two orthogonal directions of yaw and roll. Each motion branch contains a piezoelectric driver, which is responsible for outputting linear displacement and is connected to the moving bump through a spherical fit. While transmitting the displacement of the piezoelectric driver, the moving bump is allowed to offset on the spherical surface to provide additional rotational degrees of freedom. The moving bump is connected to the transmission connecting rod through a cylindrical surface fit to transmit the linear displacement of the driver to one end of the transmission connecting rod. The transmission connecting rod is connected to the rotating base through the cylindrical shaft hole as a fulcrum, and the mating plane at the other end is in point-to-surface contact with the rolling ball on the rotating end. The fit between the transmission connecting rod and the moving bump enables the linear displacement of the driver to be amplified and converted into a swing arc at the other end of the transmission connecting rod. The overall schematic diagram of the device structure and the cross-sectional view at the rotating end are shown as follows. Figure 2and Figure 3 shown.

[0055] The degree of freedom of the kinematic chain can be calculated and analyzed by formula (1).

[0056] Step 2: Solve the posture transfer relationship of the dual-degree-of-freedom dynamic derivative test device

[0057] According to the results of degree of freedom analysis, the kinematic branch chain of the dual-degree-of-freedom dynamic derivative test device structure contains two active branches, whose piezoelectric actuators are AB and A'B', the moving bumps are BC and B'C', and the transmission links are CDE and C'D'E'. Among them, the transmission link CDE contains segments CD and DE, and the structure of the link C'D'E' is similar. The rotating end EE'FG contains segments EF, E'F and FG, where EF and E'F are perpendicular to FG.

[0058] A rectangular coordinate system Gxyz is established with point G as the origin, where Gx is the roll axis, Gy is the pitch axis, and Gz is the yaw axis. The FG segment at the end of the rotation can rotate around the Gz axis, and coincides with the roll axis Gx when the angle with Gz is 0; the EF and E'F segments can rotate around the FG segment, so points E and E' can rotate around the roll axis Gx and the yaw axis Gz relative to point G.

[0059] The fixed points D and D' are located in the coordinate system Gxyz, and the connecting rods CDE and C'D'E' are hinged to the rotating base through points D and D', respectively, and are hinged to the moving bumps BC and B'C' at C and C'; similarly, BC and B'C' are hinged to the piezoelectric actuators AB and A'B' at points B and B', respectively, and points A and A' are fixed points in the coordinate system Gxyz.

[0060] The closed-loop vector method is used to construct the posture transfer model. By defining the geometric relationship and motion constraints between these components, the motion transfer and posture change of its branches are shown in formula (2), which can be rearranged to obtain formula (3).

[0061] According to the geometric constraint relationship of the EFG segment, the posture transfer model shown in formula (4) is established, and the result of formula (3) is substituted into it to solve the angle between EF and the pitch axis: , angle between EG and roll axis Substituting into formula (5) can solve the yaw angle and roll angle.

[0062] Step 3: Construct the elastic deformation correction method of the mechanism components of the dual-degree-of-freedom dynamic derivative test device;

[0063] During the transmission process of the mechanism, the load will cause elastic deformation of the component, which will be transmitted to the next kinematic pair through the kinematic chain, thus affecting the posture mapping relationship between the driver and the rotating end. In order to simplify the calculation of the deformation of the component during the transmission process, a flexible hinge model is proposed for the compressive elastic deformation of the component. Two groups of spring units with different stiffness are arranged in the orthogonal direction on the rotating pair at the hinge position between the components to equivalently describe the compressive elastic deformation characteristics of the components at both ends of the kinematic pair. Its deformation formula is shown in Equation (6).

[0064] A flexible damping contact model is proposed for the bending deformation and mechanical damping effect of components. A spring-damper unit is set along the normal direction on the contact interface to simulate the bending elastic deformation and mechanical damping characteristics of the transmission connecting rod. The model takes into account the stress distribution and energy dissipation caused by the bending deformation of the component, and realizes the quantitative characterization of the deformation and damping characteristics of the system. The formula is described in Equation (7).

[0065] Step 4: Solve the elastically corrected dual-degree-of-freedom dynamic derivative test device mechanism posture transfer model

[0066] Since the elastic deformation of the component is equivalent to the deformation of the spring unit and spring-damper unit of the revolute joint and the point-to-surface contact joint, in the posture transfer calculation, it is only necessary to supplement the deformation of the revolute joint and the point-to-surface contact joint based on the geometric relationship and on the basis of formula (3) to obtain the elastically corrected mechanism posture transfer relationship without involving the specific deformation of the component. The corrected posture transfer model is shown in formula (8).

[0067] Compared with the traditional method, the linear elastic mechanism posture transfer modeling method of the dual-degree-of-freedom dynamic derivative test device proposed in the present invention establishes an elastic unit on the kinematic pair to correct the elastic contraction and bending of the component after being stressed; the elastic amount of the component is corrected to the posture transfer model of the dual-degree-of-freedom dynamic derivative test device mechanism to obtain the mechanism posture transfer matrix taking into account the elastic deformation. Compared with the traditional method, the present invention can effectively describe the elastic deformation of the device mechanism caused by time-varying pneumatic loads, has a concise and reliable calculation formula, and the solution process is simple and clear. It not only clarifies the posture transfer characteristics of the linear elastic mechanism, but also meets the real-time requirements of motion feedback control.

Claims

1. A linear elastic mechanism posture transfer modeling method for a dual-degree-of-freedom dynamic derivative test device, characterized in that: Here are the steps: Step 1: Based on the dual-degree-of-freedom dynamic derivative test device, describe and analyze the motion branch chain; The dual-degree-of-freedom dynamic derivative test device includes a piezoelectric driver, a transmission connecting rod, a moving bump, a rotating end and a rotating base; there are four motion branches, which are connected in parallel to the rotating end, and the structures of the motion branches are the same. A cross axis is connected between the rotating end and the rotating base, and there are rotational degrees of freedom in two orthogonal directions of yaw and roll; for any motion branch, there is a piezoelectric driver as a linear displacement output, and the piezoelectric driver and the moving bump are matched through a spherical surface. The moving bump is offset along the spherical surface while transmitting the displacement output of the piezoelectric driver, thereby providing The movable protrusion fits with the transmission connecting rod through the cylindrical surface, and transmits the linear displacement of the driver to one end of the transmission connecting rod. The transmission connecting rod is connected to the rotating base through the cylindrical shaft hole as the fulcrum of the movement. The other end of the transmission connecting rod is the matching plane, which is in point-to-point contact with the rolling ball on the rotating end. The transmission connecting rod and the movable protrusion together amplify the linear displacement of the piezoelectric driver at one end and convert it into the swing arc amount at the other end. Four motion branches are symmetrically arranged in an annular array around the rotating end, and the matching plane of the transmission connecting rod is in contact with the four rolling balls on the rotating end. The calculation formula of the motion freedom of the dual-degree-of-freedom dynamic derivative test device is as follows: DOF=6n-(5P1+4P2)+P a =6×11-(5×10+4×4)+2=2(1) Where DOF represents the degree of freedom of the dynamic derivative test device mechanism, n represents the number of components in the dynamic branch chain of the dynamic derivative test device mechanism, P1 represents the number of low-pair constraints, P2 represents the number of high-pair constraints, and P a represents the driving degree of freedom; Step 2: Solve the posture transfer relationship of the dual-degree-of-freedom dynamic derivative test device; According to the calculation results of degrees of freedom, two of the motion branches of the dual-degree-of-freedom dynamic derivative test device are active branches. The piezoelectric drivers of the two branches are AB and A'B', and the moving bumps are BC and B'C'. The transmission connecting rods are CDE and C'D'E', CDE includes CD segment and DE segment, C'D'E' includes C'D segment and C'D segment; the rotating end is EE'FG, including EF segment, E'F segment and FG segment, and EF segment and E'F segment are perpendicular to FG segment; take point G as the coordinate origin, establish a rectangular coordinate system Gxyz, where Gx is the roll axis, Gy is the pitch axis, and Gz is the yaw axis. The FG segment at the rotating end can rotate around Gz, and when the angle with Gz is 0, it is aligned with the roll axis G x coincides, segment EF and segment E'F can rotate around segment FG, so endpoints E and E' can rotate around roll axis Gx and yaw axis Gz relative to endpoint G; D and D' are two fixed points in the rectangular coordinate system Gxyz, CDE and C'D'E' are hinged to the rotating base at D and D' respectively, and hinged to BC and B'C' at C and C' respectively; BC and B'C' are hinged to AB and A'B' at B and B' respectively, and A and A' are two fixed points in the rectangular coordinate system Gxyz; According to the closed-loop vector method, the vector closed loop of the motion branch chain is obtained, and its vector closed loop equation is as follows: Among them, r E 、r E' are the coordinates of point E and point E', r A 、r A' 、r G 、r F are the coordinates of point A, point A', point G and point F, l AB , l BC , l CD , l DE , l A'B' , l B'C' , l C'D' , l D'E' , l EF , l GF The vector lengths are AB, BC, CD, DE, A'B', B'C', C'D', D'E', EF, and GF respectively; Select the kinematic branch chain AB-BC-CD-DE-EF-FG as the analysis object, decompose the vector orthogonally, and expand it to get the following formula: Among them, l AB , l BC , l CD , l DE , l EF Respectively represent the vector lengths of AB, BC, CD, DE, and EF, l GE is the projection of GE on the plane oxy, α BC , α CD , α DE , α GE Respectively represent the angles between BC, CD, DE, GE and the roll axis, x AD ,y AD is the component of AD along the roll axis and pitch axis, x GD ,y GD is the component of GD along the roll axis and pitch axis, α DE0 is the initial angle between DE and the roll axis, y E For r E Component along the pitch axis; Through the vector closed-loop equation, the displacement input-output relationship under a single motion branch chain is obtained. The A'B'-B'C-C'D'-D'E'-EF-FG branch chain is selected, and the vector closed-loop equation of the same form is established. When the output of the piezoelectric driver is known, the angle and displacement of the internal components of the branch chain are calculated; Arranging equation (2) and equation (3) yields the following equation: Among them, l A'B' , l B'C' , l C'D' , l D'E' , l E'F' Respectively represent the vector lengths of A'B', B'C', C'D', D'E', and E'F', l GE' is the projection of GE' on plane oxy, α B'C' , α C'D' , α D'E' , α GE' Respectively represent the angles between B'C', C'D', D'E', GE' and the roll axis, x A'D' ,y A'D' is the component of A'D' along the roll axis and pitch axis, x GD' ,y GD' is the component of GD' along the roll axis and pitch axis, α D'E'0 is the initial angle between D'E' and the roll axis, y E' For r E' Component along the pitch axis; Select the kinematic chain EF-EE'FG-E'F and construct the displacement transfer equation as follows: Among them, y E ,y E' r E 、r E' Component along the pitch axis, l EF , l E'F , l EG , l E'G are the vector lengths of EF, E'F, EG, and E'G, respectively, and β EF , β E'F are the angles between EF, E'F and the pitch axis, α EG , α E'G are the angles between EG and E'G and the roll axis, respectively, which can be solved by equation (3). Based on the geometric relationship, the roll angle and yaw angle are calculated as follows: In the formula, α row , γ yaw are the roll angle and yaw angle respectively, β0 is the initial angle between EF and the pitch axis, α0 is the initial angle between EG and the roll axis; Step 3: Construct the elastic deformation correction method of the mechanism components of the dual-degree-of-freedom dynamic derivative test device; When the dual-degree-of-freedom dynamic derivative test device mechanism is subjected to load during transmission, elastic deformation will occur. The deformation of the dual-degree-of-freedom dynamic derivative test device component will be transmitted to the next kinematic pair through the kinematic branch chain, thereby affecting the posture mapping relationship of the piezoelectric actuator at the rotation end. In order to simplify the calculation process of the deformation transmission of the components in the dual-degree-of-freedom dynamic derivative test device mechanism, a flexible hinge model is proposed for the compressive elastic deformation of the components. Two groups of spring units with different stiffness are set in the orthogonal direction on the rotation pair at the hinge between the components, which are used for the compressive elastic deformation of the components at both ends of the equivalent kinematic pair. The formula is expressed as follows: Among them, δx and δy are the deformations along the orthogonal directions, F x 、F y is the load along two orthogonal directions, k x , k y is the stiffness along two orthogonal directions; Aiming at the bending deformation and mechanical damping of components, a flexible damping contact model is proposed. A spring damping unit is set along the normal vector on the contact surface to be used for the bending elastic deformation and mechanical damping of the equivalent transmission connecting rod. The formula is described as follows: Among them, F z represents the load along the normal vector of the contact surface, k z 、c z is the stiffness and damping along the normal vector of the contact surface, δ z , is the deformation amount and deformation rate along the normal vector of the contact surface; Step 4: Solve the elastically corrected dual-degree-of-freedom dynamic derivative test device mechanism posture transfer model; Since the elastic deformation of the components in the dual-degree-of-freedom dynamic derivative test device is equivalent to the deformation of the flexible hinge model and the flexible damping contact model on the revolute pair and the point-to-surface contact pair, in the process of calculating the posture transfer, it is only necessary to supplement the deformation of the revolute pair and the point-to-surface contact pair on the basis of formula (3) according to the geometric relationship, that is, to obtain the elastically corrected posture transfer relationship of the dual-degree-of-freedom dynamic derivative test device without considering the specific deformation of the components; the corrected posture transfer model is as follows: Among them, δ xC , δ xD is the deformation of the revolute pair C and D along the roll axis direction, δ yC , δ yD is the deformation of the revolute joint C and D along the pitch axis, δ zE It is the deformation of point-to-surface contact pair E along the normal direction of the contact surface.

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