Kinematics analysis method for 3-RSS / S parallel equipment

Abstract

The invention relates to a 3-RSS / S parallel equipment kinematics analysis method, which comprises the following steps of S1, establishing a kinematics model of mechanism coordinates and 3-RSS / S parallel equipment, and determining a geometric constraint relationship and a motion transmission path between components; s2, deriving a pose curve equation of a parallel equipment moving platform according to the 3-RSS / S parallel equipment kinematics model; s3, according to the pose curve equation, pose curves of the movable platform in the x direction, the y direction and the z direction are obtained. According to the method, the known equipment structure parameters serve as input variables and are substituted into the pose curve equation of the equipment, so that the pose state of the parallel equipment is calculated, the actual pose of the movable platform can be calculated in real time, and the influence of each structure parameter of the parallel equipment on the movable platform is better researched.

Description

Technical Field

[0001] This invention relates to the field of kinematic analysis of parallel equipment, and more specifically, to a kinematic analysis method for 3-RSS / S parallel equipment. Background Technology

[0002] With the development of equipment manufacturing technology, parallel equipment has been widely used in precision manufacturing, aerospace, and medical devices due to its advantages such as high stiffness, strong load-bearing capacity, high motion accuracy, and good dynamic response. Kinematic analysis is the foundation of the design, control, and optimization of parallel equipment, and its core task is to establish the mapping relationship between joint inputs and end effector poses. For 3-RSS / S parallel equipment, due to the coupling relationship in its structure, kinematic modeling is quite difficult. Traditional inverse kinematic analysis methods often suffer from computational complexity and poor real-time performance, making it difficult to meet the requirements of high-precision motion control and unsuitable for single-drive parallel equipment with coupled motion degrees of freedom of a single actuator. Currently, systematic analysis of the influence of link length on the pose of the moving platform of 3-RSS / S parallel equipment is incomplete, lacking effective parameter optimization basis, which limits the application of this type of equipment in high-end scenarios such as precision positioning and attitude adjustment. Summary of the Invention

[0003] The technical problem to be solved by this invention is to provide a precise and efficient kinematic analysis method for 3-RSS / S parallel equipment, which can calculate the actual position and posture of the moving platform in real time, providing technical support for equipment design optimization and high-precision control.

[0004] The technical solution adopted by this invention to solve its technical problem is: to construct a kinematic analysis method for 3-RSS / S parallel equipment, including the following steps: S1. Establish the kinematic model of the mechanism coordinates and the 3-RSS / S parallel equipment, and determine the geometric constraint relationship and motion transmission path between each component; S2. Derive the pose curve equation of the moving platform of the parallel equipment based on the kinematic model of the 3-RSS / S parallel equipment. S3. Obtain the pose curves of the launch platform in the x, y, and z directions based on the pose curve equation.

[0005] According to the above scheme, the 3-RSS / S parallel equipment includes a servo motor, a reducer, a coupling, a drive gear, a moving platform, a workpiece support frame, and a feed platform. The drive gear adopts a four-gear rotary joint, with the middle gear rigidly connected to the output shaft of the servo motor. Torque is transmitted through meshing with the three peripheral driven gears. Each driven gear is hinged to the connecting rod via an upper ball joint. The other end of the connecting rod is connected to the moving platform via a ball joint through a lower ball joint. The moving platform is driven to complete spatial attitude adjustment through the coordinated action of multiple links. The central area of ​​the moving platform forms a ball joint structure with the bearing through an S-shaped joint, forming a passive constraint chain. This constraint chain can limit the translational freedom of the moving platform, so that the moving platform can only realize rotational motion around the x, y, and z axes.

[0006] According to the above scheme, the passive branch is simplified to a link with S-joints at both ends, and a fixed coordinate system is established. B With moving coordinate system P A is fixed to the center of the upper joint and the lower joint of the intermediate branch, respectively. i and C i They represent the first i The center of the upper joint of the root active link and the first i The center of the lower joint of the root active link, D i A is the geometric center of the driving gear rotary pair. i and D i Located in the plane of the fixed coordinate system B, C i It is located in the plane of the moving coordinate system P.

[0007] According to the above plan, the upper joint center A i exist B The position vector in the coordinate system is denoted as ( i =1,2,3), lower joint center C i exist P The position vector in the coordinate system is denoted as (i=1,2,3), position vector ( i =1,2,3) and The expression for (i=1,2,3) is as follows: (1) (2) In the formula, It is a point Relative to point The distance; and They are and The polar angles corresponding to coordinate systems B and P; It is a point Relative to point The eccentric angle.

[0008] According to the above scheme, the Euler angles of the moving platform about the x, y, and z axes are set as follows: , , The position of the moving platform Represented as: (3)

[0009] According to the above scheme, the pose of this 3-RSS / S parallel equipment moving platform can be solved by the following formula: (4) In the formula, Represented as the first i The vector of the root link, Indicated as from P coordinate system to B Rotation matrix of coordinate system Represented as the center of the inferior joint C i exist P Position vector in the coordinate system Represented as P The origin of the coordinate system is at B Position vector in the coordinate system Expressed as the length of the link, Represented as the first i The axial deformation of the connecting rod, of which P The origin of the coordinate system is at B Position vector in coordinate system Represented as: (5) In the formula, h Representing the coordinate system P With coordinate system B The perpendicular distance between the two origins.

[0010] According to the above plan, P coordinate system to B Rotation matrix of coordinate system Represented as: (6) in: (7) Then, calculate according to formula (7) and formula (6) P coordinate system to B Rotation matrix of coordinate system for: (8)

[0011] According to the above scheme, the input angular velocity is obtained. With the position of the moving platform The expression for the mapping relationship between them is: (9) (10)

[0012] According to the above scheme, the pose characteristics of the 3-RSS / S parallel equipment moving platform are determined by its rotation angles around the x, y, and z axes. , , The curves that change over time and the spatial trajectory are intuitively displayed. By substituting the structural parameters of the parallel equipment into the parameters corresponding to the pose curve equations, the pose curves of the moving platform of the 3-RSS / S parallel equipment can be obtained, which are the rotation angle curves of the moving platform in the x, y, and z directions.

[0013] According to the above scheme, in step S3, the method for optimizing the pose curve includes: adjusting the independent variable time in the pose curve equation. t Discretize the data into several uniform sampling points, and substitute them into the above solution formula to obtain the corresponding values ​​for each sampling point. , , Numerical analysis; the least squares method is used to fit curves to discrete data, generating continuous and smooth curves. , , Rotate the curve to check the smoothness and accuracy of the fitted curve and remove outlier data points.

[0014] The kinematic analysis method for 3-RSS / S parallel equipment of the present invention has the following beneficial effects: 1. This invention uses known equipment structural parameters as input variables and substitutes these parameters into the equipment's pose curve equation to calculate the pose state of the parallel equipment. This allows for real-time estimation of the actual pose of the moving platform, thus enabling a better study of the influence of various structural parameters of the parallel equipment on the moving platform.

[0015] 2. This invention solves the position and attitude equations of the moving platform of the 3-RSS / S parallel equipment. At the same time, based on the derived kinematic formula of the moving platform of the 3-RSS / S parallel equipment, the influence of the link length of the 3-RSS / S parallel equipment on the rotation angle of the moving platform is investigated. Attached Figure Description

[0016] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings: Figure 1 This is a structural schematic diagram of a 3-RSS / S parallel equipment; Figure 2 The 3-RSS / S diagram is a detailed structural configuration and working principle diagram of the PKM. Figure 3 This is a simplified kinematic model diagram of PKM with 3-RSS / S. Figure 4 This is the pose curve of the moving platform when l=199mm; Figure 5 This is the pose curve of the moving platform when l=200mm; Figure 6 This is the pose curve of the moving platform when l=205mm; Figure 7 It is a spatial trajectory diagram of the 3-RSS / S parallel equipment moving platform; Figure 8 This is a schematic diagram comparing the pose curves of different link lengths in a 3-RSS / S parallel equipment. Detailed Implementation

[0017] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0018] The kinematic analysis method for 3-RSS / S parallel equipment of the present invention includes the following steps: S1. Establish the kinematic model of the mechanism coordinates and the 3-RSS / S parallel equipment, clarify the geometric constraint relationship and motion transmission path between each component, and lay the foundation for the calculation of the positive motion posture.

[0019] Figure 1 The overall configuration of the 3-RSS / S parallel equipment was demonstrated. Figure 2 The detailed structural configuration and working principle of the equipment are further provided. This parallel equipment adopts a 3-RSS / S structure, and the entire system consists of key functional components such as a servo motor, reducer, coupling, drive gear, moving platform, workpiece support frame, and feed platform. The drive module uses a four-gear revolute joint design, with the middle gear rigidly connected to the servo motor output shaft. Precise torque transmission is achieved through meshing with the three surrounding driven gears. Each driven gear is hinged to the connecting rod via an upper ball joint, driving the connecting rod to perform controllable motion. The other end of the connecting rod is connected to the moving platform via a lower ball joint, and the moving platform completes spatial attitude adjustment through the coordinated action of multiple links. The central area of ​​the moving platform forms a ball joint structure with GX bearings via an S-joint, forming the passive constraint chain of the equipment. This constraint mechanism effectively restricts the translational freedom of the moving platform, allowing it to only achieve rotational motion around the x, y, and z axes, ensuring the motion accuracy and stability of the equipment.

[0020] Figure 3This is a simplified kinematic model of a 3-RSS / S PKM, where the passive branches are simplified to links with S-joints at both ends. (Fixed coordinate system) B With moving coordinate system P They are respectively fixed to the center of the upper and lower joints of the intermediate branch chain. A i and C i They represent the first i The center of the upper and lower joints of the root active link, D i A is the geometric center of the driving gear rotary joint (R joint). i and D i Located in the plane of the fixed coordinate system B, C i It is located in the plane of the moving coordinate system P.

[0021] S2. Based on the kinematic model of the 3-RSS / S parallel equipment, derive the pose curve equation of the moving platform of the parallel equipment.

[0022] In the coordinate system of the 3-RSS / S parallel equipment established above, the upper joint center A is... i exist B The position vector in the coordinate system is denoted as ( i =1,2,3), lower joint center C i exist P The position vector in the coordinate system is denoted as (i=1,2,3), then the position vector ( i =1,2,3) and The expression (i=1,2,3) can be represented as: (1) (2) In the above position vector expression, It is a point Relative to point The distance. and They are and The polar angles corresponding to coordinate systems B and P. It is a point Relative to point The eccentric angle.

[0023] To obtain the pose curve equation of the 3-RSS / S parallel equipment moving platform, it is necessary to obtain the rotation angles of the moving platform about the x, y, and z axes respectively. The Euler angles of the moving platform about the x, y, and z axes are now set as follows: , , The position of the moving platform It can be represented as: (3) During the operation of the 3-RSS / S parallel equipment, since the model only considers the axial deformation of the connecting rod and not its lateral deformation, the pose of the moving platform of the 3-RSS / S parallel equipment can be solved in the following way: (4) In solving the equation, Represented as the first i The vector of the root link, Indicated as from P coordinate system to B Rotation matrix of coordinate system Represented as the center of the inferior joint C i exist P Position vector in the coordinate system Represented as P The origin of the coordinate system is at B Position vector in the coordinate system Expressed as the length of the link, Represented as the first i The axial deformation of the connecting rod can be measured using a laser displacement sensor. P The origin of the coordinate system is at B Position vector in coordinate system It can be represented as: (5) in, h Representing the coordinate system P With coordinate system B The perpendicular distance between the two origins. P coordinate system to B Rotation matrix of coordinate system It can be represented as: (6) in: (7) Then, according to formulas (7) and (6), we can calculate... P coordinate system to B Rotation matrix of coordinate system for: (8) Substituting the above-derived formula into the 3-RSS / S parallel equipment moving platform pose calculation formula (4), the input angular velocity can be obtained. With the position of the moving platform The mapping relationship between them can be expressed as follows: (9) In the expression , , They are respectively: .

[0025] S3. Based on the pose curve equation of the 3-RSS / S parallel equipment, calculate the pose curve of the moving platform in the x, y, and z directions.

[0026] The pose characteristics of the 3-RSS / S parallel equipment moving platform can be determined by its rotation angles around the x, y, and z axes. , , The curves and spatial trajectories that change over time are visually represented, and the link length is also shown. It is one of the key parameters affecting the pose motion of the moving platform of the 3-RSS / S parallel equipment. Combining the kinematic equations of the 3-RSS / S parallel equipment derived above, the pose curves of the moving platform in the x, y, and z directions can be calculated.

[0027] In the expression, Expressed as the length of the link, Represented as the first i The axial deformation of the connecting rod. Indicates the radius of the center of the lower joint of the connecting rod. Represented as the center polar radius of the drive gear. It is expressed as the distance between the center of the joint on the connecting rod and the center of the gear shaft. It is represented as the polar angle at the center of the lower joint. Represented as the polar angle of the geometric center of the driving gear. It is expressed as the eccentricity angle of the upper joint center relative to the geometric center of the drive gear. h This represents the distance between the origins of the static and dynamic coordinate systems. Then, by substituting the structural parameters of the parallel equipment into the corresponding parameters of the pose curve equation, the pose curve of the moving platform of the 3-RSS / S parallel equipment can be obtained, which is the rotation angle curve of the moving platform in the x, y, and z directions.

[0028] Finally, optimize the pose curve, and add time as the independent variable in the pose curve equation. t Discretize the data into several uniform sampling points, and substitute them into the above solution formula to obtain the corresponding values ​​for each sampling point. , , Numerical analysis; the least squares method is used to fit curves to discrete data, generating continuous and smooth curves. , , The rotation angle curve is used to verify the smoothness and accuracy of the fitted curve, removing outlier data points to ensure the curve matches the actual motion of the equipment. A mathematical model is used to smooth the discrete observation data, allowing for a more intuitive representation of the motion of the 3-RSS / S parallel equipment platform. Smoothness verification ensures the pose curve is physically reasonable, meaning the equipment's motion is continuous and without abrupt changes, conforming to dynamic principles and avoiding unrealistic trajectories. Accuracy verification ensures the deviation between the curve and the original observation data points is within an acceptable range, guaranteeing the curve's accuracy. If the smoothness and accuracy verification are inadequate, the pose curve may be unreliable, potentially deviating significantly from the actual motion trajectory, losing its predictive and guiding significance, and thus failing to accurately predict the equipment's position and attitude at a future point in time. During the calculation process, outlier data points may occur, primarily due to calculation errors, sensor malfunctions, or human error. Since these outliers do not reflect the true motion of the equipment, retaining them would interfere with the curve fitting process and severely affect the accuracy of the pose curve; therefore, these outlier data points must be removed.

[0029] The present invention also provides a specific example as follows: Taking 3-RSS / S parallel equipment as the research object, based on a fixed coordinate system B With moving coordinate system P The settings (fixed at the centers of the upper and lower joints of the intermediate branch respectively) are used to establish the mechanism coordinate system. Among them, A... i (i=1,2,3) is the i-th i C, the center of the joint on the root active link i (i=1,2,3) is the i-th i Root active link lower joint center, D i To ensure A is the geometric center of the driving gear rotary pair i and D i Located in a fixed coordinate system B The plane in which it is located, C i Located in a moving coordinate system P The model considers only the axial elastic deformation of the connecting rod and ignores the lateral deformation. By simplifying the system's degrees of freedom, the computational complexity is reduced. Key mathematical representations such as the center position vectors and rotation matrices of each joint are clearly defined, forming a complete simplified kinematic model of the 3-RSS / S parallel equipment.

[0030] First, based on the structure diagram of the 3-RSS / S parallel equipment, the input angular velocity of the moving platform of the 3-RSS / S parallel equipment is obtained. With the position of the moving platform Mapping relationship between them:

[0031] in,

[0032] The structural parameters involved in the above formulas are shown in Table 1. Substituting the structural parameters into the formulas will yield the pose of the moving platform. .

[0033] Table 1 Structural Parameters

[0034] Substituting the link lengths of 199mm, 200mm, and 205mm into the pose curve equation, respectively, the rotation angles of the moving platform about the x, y, and z axes are solved using a calculation program. , , A curve that changes over time. Record. and The range of fluctuations and The absolute value. From the experimental results ( Figure 4 , 5 6) It can be seen that the rotation angle of the moving platform on the x, y, and z axes is... , , All exhibit a periodic fluctuation pattern, but the fluctuation amplitude varies with the length of the connecting rod. Increased in size and significantly improved, for example, when the connecting rod length is 199mm. , The fluctuation range is only about ±1°, while when the connecting rod length is 205mm... , The fluctuation range can reach ±3°, while the rotation angle on the z-axis is... Its absolute value increases with the increase of the connecting rod length. When the connecting rod length is 199mm, Approximately 6.94°, when the connecting rod length is 205mm. The angle was raised to approximately 23.7°. Based on the experimental results, the influence of link length on the pose of the moving platform was clarified. In practical applications, a suitable link length can be selected according to the specific needs of precision positioning and attitude adjustment.

[0035] Meanwhile, the spatial trajectory diagram further illustrates the pose distribution corresponding to different link lengths, such as Figure 7 As shown, the position trajectory of the moving platform is elliptical for all three link lengths. However, as the link length increases, its rotation angle around the z-axis increases. An increase in absolute value is reflected as an upward shift of the ellipse on the pose trajectory, directly demonstrating the direct control effect of the link length on the rotation range of the moving platform. Meanwhile, as... Figure 8The comparison diagram of pose curves for three link lengths shows that as the link length increases, its rotation angles around the x-axis and y-axis also increase. , The fluctuation range increased significantly, and the rotation angle around the z-axis also increased. The amplitude increases. This characteristic indicates that by adjusting the link length of the 3-RSS / S parallel mechanism, the pose range of the moving platform can be flexibly changed, providing a basis for parameter optimization in its application in precision positioning and attitude adjustment scenarios.

[0036] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A kinematic analysis method for 3-RSS / S parallel equipment, characterized in that, Includes the following steps: S1. Establish the kinematic model of the mechanism coordinates and the 3-RSS / S parallel equipment, and determine the geometric constraint relationship and motion transmission path between each component; S2. Derive the pose curve equation of the moving platform of the parallel equipment based on the kinematic model of the 3-RSS / S parallel equipment. S3. Obtain the pose curves of the launch platform in the x, y, and z directions based on the pose curve equation.

2. The kinematic analysis method for 3-RSS / S parallel equipment according to claim 1, characterized in that, The 3-RSS / S parallel equipment includes a servo motor, a reducer, a coupling, a drive gear, a moving platform, a workpiece support frame, and a feed platform. The drive gear adopts a four-gear rotary joint, with the middle gear rigidly connected to the output shaft of the servo motor. Torque is transmitted through meshing with the three peripheral driven gears. Each driven gear is hinged to the connecting rod via an upper ball joint. The other end of the connecting rod is connected to the moving platform via a ball joint through a lower ball joint. The moving platform is driven to complete spatial attitude adjustment through the coordinated action of multiple links. The central area of ​​the moving platform forms a ball joint structure with the bearing through an S-shaped joint, forming a passive constraint chain. This constraint chain can limit the translational freedom of the moving platform, so that the moving platform can only realize rotational motion around the x, y, and z axes.

3. The kinematic analysis method for 3-RSS / S parallel equipment according to claim 2, characterized in that, In step S1, the passive branch is simplified to a link with S-joints at both ends, and a fixed coordinate system is established. B With moving coordinate system P A is fixed to the center of the upper joint and the lower joint of the intermediate branch, respectively. i and C i They represent the first i The center of the upper joint of the root active link and the first i The center of the lower joint of the root active link, D i A is the geometric center of the driving gear rotary pair. i and D i Located in the plane of the fixed coordinate system B, C i It is located in the plane of the moving coordinate system P.

4. The kinematic analysis method for 3-RSS / S parallel equipment according to claim 3, characterized in that, In step S2, the upper joint center A i exist B The position vector in the coordinate system is denoted as ( i =1,2,3), lower joint center C i exist P The position vector in the coordinate system is denoted as (i=1,2,3), position vector ( i =1,2,3) and The expression for (i=1,2,3) is as follows: （1） （2） In the formula, It is a point Relative to point The distance; and They are and The polar angles corresponding to coordinate systems B and P; It is a point Relative to point The eccentric angle.

5. The kinematic analysis method for 3-RSS / S parallel equipment according to claim 4, characterized in that, In step S2, the Euler angles of the moving platform about the x, y, and z axes are set as follows: , , The position of the moving platform Represented as: （3）。 6. The kinematic analysis method for 3-RSS / S parallel equipment according to claim 5, characterized in that, In step S2, the pose of this 3-RSS / S parallel equipment moving platform is solved by the following formula: （4） In the formula, Represented as the first i The vector of the root link, Indicated as from P coordinate system to B Rotation matrix of coordinate system Represented as the center of the inferior joint C i exist P Position vector in the coordinate system Represented as P The origin of the coordinate system is at B Position vector in the coordinate system Expressed as the length of the link, Represented as the first i The axial deformation of the connecting rod, of which P The origin of the coordinate system is at B Position vector in coordinate system Represented as: （5） In the formula, h Representing the coordinate system P With coordinate system B The perpendicular distance between the two origins.

7. The kinematic analysis method for 3-RSS / S parallel equipment according to claim 6, characterized in that, In step S2, P coordinate system to B Rotation matrix of coordinate system Represented as: （6） in: （7） Then, calculate according to formula (7) and formula (6) P coordinate system to B Rotation matrix of coordinate system for: （8）。 8. The kinematic analysis method for 3-RSS / S parallel equipment according to claim 7, characterized in that, In step S2, the input angular velocity is obtained. With the position of the moving platform The expression for the mapping relationship between them is: （9） （10）。 9. The kinematic analysis method for 3-RSS / S parallel equipment according to claim 8, characterized in that, In step S3, the pose characteristics of the 3-RSS / S parallel equipment moving platform are determined by its rotation angles around the x, y, and z axes. , , The curves that change over time and the spatial trajectory are intuitively displayed. By substituting the structural parameters of the parallel equipment into the parameters corresponding to the pose curve equations, the pose curves of the moving platform of the 3-RSS / S parallel equipment can be obtained, which are the rotation angle curves of the moving platform in the x, y, and z directions.

10. The kinematic analysis method for 3-RSS / S parallel equipment according to claim 8, characterized in that, In step S3, the method for optimizing the pose curve includes: adjusting the independent variable time in the pose curve equation. t Discretize the data into several uniform sampling points, and substitute them into the above solution formula to obtain the corresponding values ​​for each sampling point. , , Numerical analysis; the least squares method is used to fit curves to discrete data, generating continuous and smooth curves. , , Rotate the curve to check the smoothness and accuracy of the fitted curve and remove outlier data points.

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