Design method of parallel mechanism with both high normal stiffness and high lateral stiffness

Abstract

The present invention relates to a design method for a parallel mechanism with both high normal stiffness and high lateral stiffness, comprising the following steps: S1, determining the motion freedom requirements of the parallel mechanism dynamic platform, establishing the mechanism motion spinor system and constraint spinor system, and establishing the branch constraint spinor system and motion spinor system of the parallel mechanism; S2, establishing a force transmission model of the parallel mechanism, and determining the maximum design criterion for normal stiffness; S3, establishing the force balance equation of the parallel mechanism, introducing an auxiliary configuration to provide an additional force system to adjust the overall stress state of the mechanism, and determining the design criterion for maximizing lateral stiffness; S4, combining the maximum design criterion for normal stiffness and the design criterion for maximizing lateral stiffness with the dynamic and static force model to obtain a parallel mechanism with both high normal stiffness and high lateral stiffness. The present invention establishes a design criterion for maximum normal stiffness of the parallel mechanism to maximize the normal stiffness of the mechanism, and has important application prospects in parallel mechanisms under heavy load conditions.

Description

Technical Field

[0001] The present invention relates to the field of parallel mechanism configuration design, and more particularly to a design method for a parallel mechanism with both high normal stiffness and high lateral stiffness. Background Art

[0002] Parallel mechanisms are widely used in handling, packaging, forming and other fields due to their advantages such as strong load-bearing capacity, high rigidity and high precision. The primary task of parallel mechanism research and development is to design a configuration that meets the requirements of motion freedom / motion trajectory. At present, a variety of synthesis methods have been proposed by scholars, mainly including those based on screw theory, displacement group theory, virtual loop method and POC set method. However, the stiffness performance of the parallel mechanism synthesized by the above methods is difficult to be effectively guaranteed. The low rigidity mechanism leads to reduced forming accuracy of the mechanism, especially under heavy load conditions in the forming manufacturing field. With the increasing demand for lightweight, large-scale, extreme structures, high temperature resistance and high load-bearing components, greater forming force and higher rigidity are required for parallel mechanism forming equipment. The main and lateral rigidity of existing parallel mechanisms are low, resulting in poor stability of the equipment structure and low accuracy of movement. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a design method for a parallel mechanism with both high normal stiffness and high lateral stiffness, establish a design criterion for the maximum normal stiffness of the parallel mechanism, so as to maximize the normal stiffness of the mechanism, and have important application prospects in parallel mechanisms under heavy load conditions.

[0004] The technical solution adopted by the present invention to solve the technical problem is: constructing a design method for a parallel mechanism with both high normal stiffness and high lateral stiffness, comprising the following steps:

[0005] S1. Determine the required degrees of freedom of motion of the parallel mechanism's dynamic platform, establish the mechanism's motion screw system and constraint screw system, and on this basis, establish the parallel mechanism's branch constraint screw system and motion screw system, and select the motion branch and mechanism configuration that meet the parallel mechanism's design constraints and motion requirements;

[0006] S2. Establish a force transmission model for the parallel mechanism and determine the maximum design criterion for normal stiffness;

[0007] S3. Establish the force balance equation of the parallel mechanism, introduce auxiliary configurations to provide additional force systems to adjust the overall force state of the mechanism, and determine the design criteria for maximizing lateral stiffness;

[0008] S4. Combining the design criteria of maximum normal stiffness and maximum lateral stiffness with the static and dynamic force models, a parallel mechanism with both high normal stiffness and high lateral stiffness is obtained.

[0009] According to the above scheme, in step S1, arbitrary spatial motion is characterized by a six-dimensional motion spinor:

[0010]

[0011] Where, The system of motion spinors representing spatial motion, S mpi Represents the orthogonal spinor of spatial motion; based on the spinor theory, the relationship between the spinor system of the parallel mechanism motion platform and the spinor of the branch joint motion is expressed as:

[0012]

[0013] Where, represents a constrained screw system that is reciprocal with the moving platform's motion screw system, represents the constrained screw system of the parallel mechanism branch, The motion rotation system of the parallel mechanism branch chain; various types of branches are generated using the linear combination of the branch chain motion rotations, and the parallel mechanism that meets the motion requirements is obtained through system assembly and drive design.

[0014] According to the above scheme, in step S2, the total static balance of the parallel mechanism is expressed as:

[0015]

[0016] in, It represents the constraint force of the platform on the driving joint. represents the drive provided by the driven joint, Σ represents the angle of the base guide, and F n represents the normal force; the overall static equilibrium of the parallel mechanism can be expressed as:

[0017]

[0018] in, Represents the components of the connecting rod along the x, y, and z directions, J g The Jacobian matrix of the connecting rod force transmission of the parallel mechanism is represented by:

[0019] Principle 1:

[0020] According to the above solution, in step S3, by designating the point of action of the external force as the force analysis origin, when the parallel mechanism is acted upon by the lateral force, the force system of the parallel mechanism is expressed as:

[0021]

[0022] Among them, ξ r The coefficient representing the radial force, ξt The coefficient representing the tangential force, the driving force provided by each branch chain is expressed as follows:

[0023]

[0024] in, represents the connecting rod torque, Represents the three components of the connecting rod lever arm; the force balance condition is:

[0025]

[0026] because, The force balance in the z-axis is expressed as:

[0027]

[0028] In order to minimize the connecting rod force, the optimization goal is:

[0029]

[0030] Another parallel mechanism is introduced to avoid the motion singularity and withstand the lateral force. The force balance condition of the other parallel mechanism is:

[0031]

[0032] in, The sub-table represents the vector components of the hydraulic connecting rod, Express the forces of the three hydraulic connecting rods and solve them by the following equation:

[0033]

[0034] Among them, J h is the force transmission Jacobian matrix of the two parallel mechanisms; in order to maximize the lateral stiffness, the design criteria of the other parallel mechanism are as follows:

[0035] Design Guidelines:

[0036] Design goals:

[0037]

[0038] According to the above solution, in step S4, the torque of the six connecting rods and the torque of the hydraulic connecting rod are calculated as follows:

[0039]

[0040] in, and is the force torque and The unit spinor, r l i and Represents the direction vector of the connecting rod and the hydraulic connecting rod, d l0 Indicates the length of the connecting rod, d h0 and They represent the initial length and extension length of the hydraulic connecting rod respectively; the static equilibrium equation is expressed as follows:

[0041]

[0042] Among them, J g is the connecting rod force Jacobian matrix, J h is the hydraulic connecting rod force Jacobian matrix.

[0043] The driving force vector is expressed as:

[0044]

[0045] The parameter combination of the two parallel mechanisms becomes the following optimization problem:

[0046]

[0047] The design method of the parallel mechanism with both high normal stiffness and high lateral stiffness according to the present invention has the following beneficial effects:

[0048] This invention establishes a design criterion for the maximum normal stiffness of a parallel mechanism to maximize its normal stiffness, resulting in a near-singular 6-PSS parallel mechanism with maximum normal stiffness. Furthermore, a new auxiliary 3-SPS configuration is introduced to provide an additional force system to maximize the lateral stiffness of the mechanism, and a design criterion for maximizing lateral stiffness is proposed, namely, that all SS link forces should maintain positive values. This invention provides a new approach for synthesizing heavy-duty parallel mechanisms with high normal and high lateral stiffness, and has important application prospects in parallel mechanisms operating under heavy-duty conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] The present invention will be further described below with reference to the accompanying drawings and embodiments, in which:

[0050] Figure 1 It is a basic structural diagram of the 6-PSS / 3-SPS parallel mechanism;

[0051] Figure 2 It is a schematic diagram of the static model of the 6-PSS / 3-SPS parallel mechanism;

[0052] Figure 3 Schematic diagram of the comprehensive error of the dynamic platform of the 6-PSS / 3-SPS parallel mechanism under the maximum normal force;

[0053] Figure 4 Schematic diagram of the comprehensive error of the dynamic platform of the 6-PSS / 3-SPS parallel mechanism under different normal forces;

[0054] Figure 5 It is a schematic diagram of the normal stiffness of the 6-PSS / 3-SPS parallel mechanism;

[0055] Figure 6 Schematic diagram of the error of the dynamic platform of the 6-PSS parallel mechanism under different lateral forces;

[0056] Figure 7 Schematic diagram of the moving platform error of the 6-PSS / 3-SPS parallel mechanism under different lateral forces;

[0057] Figure 8 Schematic diagram of the lateral stiffness of the 6-PSS and 6-PSS / 3-SPS parallel mechanisms. DETAILED DESCRIPTION

[0058] In order to have a clearer understanding of the technical features, purposes and effects of the present invention, specific embodiments of the present invention are now described in detail with reference to the accompanying drawings.

[0059] The design method of the parallel mechanism with both high normal stiffness and high lateral stiffness of the present invention comprises the following steps:

[0060] S1. According to the required spatial motion, determine the required degrees of freedom of motion of the parallel mechanism dynamic platform, establish the mechanism motion screw system and constraint screw system, and on this basis establish the branch constraint screw system and motion screw system of the parallel mechanism. Arbitrary spatial motion can be represented by six-dimensional motion screw:

[0061]

[0062] Where, The system of motion spinors representing spatial motion, S mpi represents the orthogonal spinor of spatial motion. Obviously, the rank of the spinor system of spatial motion is the degree of freedom of the required parallel mechanism. Based on the spinor theory, the relationship between the spinor system of the parallel mechanism motion platform and the spinor of the branch joint motion can be expressed as:

[0063]

[0064] Where, represents a constrained screw system that is reciprocal with the moving platform's motion screw system, represents the constrained screw system of the parallel mechanism branch, The kinematic rotation system of the parallel mechanism's branches. Based on the above process, various types of branches can be generated using linear combinations of the branch's kinematic rotations. Subsequently, through system assembly and drive design, a wide range of parallel mechanisms that meet the kinematic requirements can be obtained, as shown in Table 1. In this table, P represents a translational joint, R represents a revolute joint, U represents a universal joint, S represents a spherical joint, and C represents a cylindrical joint.

[0065] Table 1 Typical joint layout between the driving joint and the moving platform.

[0066]

[0067]

[0068] For heavy-duty operating conditions, it is best to minimize the number of joints. Therefore, the joint configuration of the branch with the fewest joints can be SU, US, or CS, with only two joints per branch. Furthermore, for ease of manufacturing, variations in joint type should be minimized. U-joints generally consist of two rotating shafts connected by a cross axis, resulting in a complex structure and low stiffness. S-joints, on the other hand, can be designed as a contact of spherical parts, simplifying the structure and increasing stiffness. Therefore, the U-joints of the US and SU branches can be configured as SS. This arrangement allows for the freedom of both branches. From a kinematic perspective, the rotation of the connecting rod along the axis does not affect the structure, making the SS joint layout feasible. From a static and dynamic perspective, the deformation of the connecting rod is caused by the forces at the joints at both ends. For SS and US nodes, the total force acting on both ends of the node is the same, resulting in no significant difference in deformation. Therefore, the 6-PSS or 6-RSS parallel mechanism offers unique advantages under heavy-duty operating conditions.

[0069] S2. Establish a force transmission model for the parallel mechanism and propose a design criterion for the maximum normal stiffness. The total static equilibrium of the 6-PSS parallel mechanism can be expressed as:

[0070]

[0071] in, It represents the constraint force of the platform on the driving joint. represents the drive provided by the driven joint, Σ represents the angle of the base guide, and F n Indicates the normal force. Considering that the normal force on the mechanism is large under heavy load, when Σ approaches 0, The smaller it is, the greater the system stiffness is, which means that the driving force along the horizontal direction is optimal. Furthermore, the overall static balance of 6-PSS can be expressed as:

[0072]

[0073] in, Represents the components of the connecting rod along the x, y, and z directions, J g The Jacobian matrix of the connecting rod force transmission of the 6-PSS parallel mechanism is represented by n acts mainly in the vertical direction, so the connecting rod force It should also be in the vertical direction to minimize the connecting rod force. This means that the horizontal component of the connecting rod force and It should be minimized as much as possible. Therefore, the design criterion for maximizing the normal stiffness of the 6-PSS parallel mechanism is:

[0074] Principle 1:

[0075] S3. The force balance equations for the parallel mechanism are derived. By introducing an auxiliary configuration to provide an additional force system to adjust the overall force state of the mechanism, a design criterion for maximizing lateral stiffness is proposed accordingly. Based on the design criterion of S2, the 6-PSS parallel mechanism is well suited to withstand normal forces, but balancing lateral forces is challenging. Furthermore, by specifying the point of application of the external force as the force analysis origin, the force system of the 6-PSS mechanism when the parallel mechanism is subjected to lateral forces can be expressed as:

[0076]

[0077] Among them, ξ r The coefficient representing the radial force, ξ t The coefficient representing the tangential force, the driving force provided by each branch chain is expressed as follows:

[0078]

[0079] in, represents the connecting rod torque, represents the three components of the connecting rod lever arm; the force balance condition can be further written as:

[0080]

[0081] because, Therefore, the force balance in the z-axis can be expressed as:

[0082]

[0083] In order to minimize the connecting rod force, the optimization goal is:

[0084]

[0085] As can be seen from Equation (12), in order to minimize the value of the equation, all terms should be greater than zero; that is, all link forces are greater than 0. This provides a design criterion for improving lateral stiffness. Therefore, all terms in Equation (12) are greater than zero, and the contribution of each term is positive, that is, the work of each link is positive. This illustrates the mechanism of enhancing stiffness by making the links perform positive work, which has never been disclosed before. However, for this almost singular 6-PSS parallel mechanism configuration, it is difficult to achieve force balance. Therefore, a 3-SPS parallel mechanism is introduced to circumvent the motion singularity and withstand the lateral force. In order to reduce the complexity of control, a flexible hydraulic link is used to balance the lateral force and cooperate with the 6-PSS parallel mechanism to adjust the position error.

[0086] The force balance condition of the new parallel mechanism will be:

[0087]

[0088] in, The sub-table represents the vector components of the hydraulic connecting rod, The forces of the three hydraulic connecting rods can be solved by the following equation:

[0089]

[0090] Among them, J h is the force transmission Jacobian matrix of the 6-PSS / 3-SPS parallel mechanism. In order to maximize the lateral stiffness, the design criteria of the 3-SPS parallel mechanism are as follows:

[0091]

[0092] The purpose of adding a 3-SPS parallel mechanism is to maintain the positive link force of the 6-PSS parallel mechanism while achieving lateral force balance by actively adjusting the hydraulic pressure. The hydraulic pressure should be neither too large nor too small, as either extreme would disrupt the balance. However, there are many configurations, so the design goal is to minimize the hydraulic link force.

[0093] S4. Combining the above design criteria with the static model, a near-singular 6-PSS / 3-SPS parallel mechanism with high primary and lateral stiffness is obtained. The force rotation of the six connecting rods and the force rotation of the hydraulic connecting rod can be calculated as follows:

[0094]

[0095] in, and is the force torque and The unit spinor, r l iand Represents the direction vector of the connecting rod and the hydraulic connecting rod, d l0 Indicates the length of the connecting rod, d h0 and They represent the initial length and extension length of the hydraulic connecting rod respectively. The static equilibrium equation can be expressed as follows:

[0096]

[0097] Among them, J g is the connecting rod force Jacobian matrix, J h is the hydraulic connecting rod force Jacobian matrix.

[0098] The driving force vector can be expressed as:

[0099]

[0100] This shows that if the external force and the force applied by the hydraulic linkage are known, the magnitude of the driving force vector can be determined.

[0101] The parameter synthesis of the 6-PSS / 3SPS parallel mechanism becomes the following optimization problem:

[0102]

[0103] Although the optimization goal is to minimize the root mean square error of the horizontal projection and hydraulic pressure of all links, this is not directly related to high stiffness. According to the design criteria proposed in S2 and S3, the design criteria is to maximize both normal and lateral stiffness. Therefore, the physical meaning of the optimization goal can be seen as maximizing both normal and lateral stiffness.

[0104] Example

[0105] Combining the above-mentioned design method of a six-DOF parallel mechanism with high main and lateral stiffness, a flexible redundant 6-PSS / 3-SPS parallel mechanism with high normal stiffness and lateral stiffness is designed. The design parameters are shown in Table 3. The basic structure is shown in Figure 1 As shown, the static model is as Figure 2 shown.

[0106] Table 3 Design parameters of the 6-PSS / 3-SPS parallel mechanism.

[0107]

[0108] According to the design parameters, the normal and lateral stiffness performance of the new 6-PSS / 3-SPS parallel mechanism is analyzed. First, in order to evaluate the normal stiffness performance of the new 6-PSS / 3-SPS, the normal external force is set to F n =4~8MN, the lever arm is R e=45~90mm, lateral force is 0, and the maximum error of the moving platform is calculated, such as Figure 3 As shown in the figure, the dynamic platform error under different normal forces and different force arms is further analyzed, as shown in the figure. Figure 4 As shown; Figure 5 The normal stiffness of the 6-PSS / 3-SPS under different force arms and normal forces is given. It is observed that the normal stiffness decreases with increasing force action radius, while the change of normal force has no significant effect on it. The maximum stiffness of the 6-PSS / 3-SPS mechanism is very high, ranging from about 21.5N / nm to 29.2N / nm, and is not affected by the change of normal force. Furthermore, in order to evaluate the lateral stiffness performance of the new 6-PSS / 3-SPS, the lateral force coefficient is designed to be ξ r =0~0.1 and ξ t =0~0.1. Figure 6 The error distribution of the 6-PSS parallel mechanism dynamic platform under different radial and tangential forces is described, and the error distribution ranges from 245 to 809 μm. Figure 7 The error distribution of the 6-PSS / 3-SPS parallel mechanism dynamic platform under different radial and tangential forces is described, and the error distribution ranges from 207 to 454 μm. Figure 8 The lateral stiffness of the 6-PSS and 6-PSS / 3-SPS parallel mechanisms is summarized. It can be seen that the lateral stiffness of the 6-PSS increases from approximately 0.21 N / nm to 2.1 N / nm, while the lateral stiffness of the 6-PSS / 3-SPS parallel mechanism increases from 1.3 N / nm to 12.9 N / nm, an increase of approximately six times. In summary, the proposed 6-PSS / 3-SPS parallel mechanism design has high normal stiffness and is suitable for heavy-load conditions.

[0109] The embodiments of the present invention are described above in conjunction with the accompanying drawings, but the present invention is not limited to the above-mentioned specific implementation methods. The above-mentioned specific implementation methods are merely illustrative and not restrictive. Under the guidance of the present invention, ordinary technicians in this field can also make many forms without departing from the scope of protection of the present invention and the claims, all of which are protected by the present invention.

Claims

1. A design method for a parallel mechanism with both high normal stiffness and high lateral stiffness, characterized in that: The following steps are involved: S1. Determine the required degrees of freedom of motion of the parallel mechanism's dynamic platform, establish the mechanism's motion screw system and constraint screw system, and on this basis, establish the parallel mechanism's branch constraint screw system and motion screw system, and select the motion branch and mechanism configuration that meet the parallel mechanism's design constraints and motion requirements; S2. Establish a force transmission model for the parallel mechanism and determine the maximum design criterion for normal stiffness; S3. Establish the force balance equation of the parallel mechanism, introduce auxiliary configurations to provide additional force systems to adjust the overall force state of the mechanism, and determine the design criteria for maximizing lateral stiffness; S4. Combining the design criteria of maximum normal stiffness and maximum lateral stiffness with the static and dynamic force models, a parallel mechanism with both high normal stiffness and high lateral stiffness is obtained.

2. The design method of a parallel mechanism with both high normal stiffness and high lateral stiffness according to claim 1, characterized in that: In step S1, arbitrary spatial motion is characterized by a six-dimensional motion spinor: Where, The system of motion spinors representing spatial motion, S mpi Represents the orthogonal spinor of spatial motion; based on the spinor theory, the relationship between the spinor system of the parallel mechanism motion platform and the spinor of the branch joint motion is expressed as: Where, represents a constrained screw system that is reciprocal with the moving platform's motion screw system, represents the constrained screw system of the parallel mechanism branch, The motion rotation system of the parallel mechanism branch chain; various types of branches are generated using the linear combination of the branch chain motion rotations, and the parallel mechanism that meets the motion requirements is obtained through system assembly and drive design.

3. The design method of a parallel mechanism with both high normal stiffness and high lateral stiffness according to claim 2, characterized in that: In step S2, the total static balance of the parallel mechanism is expressed as: in, It represents the constraint force of the platform on the driving joint. represents the drive provided by the driven joint, Σ represents the angle of the base guide, and F n represents the normal force; the overall static equilibrium of the parallel mechanism can be expressed as: in, Represents the components of the connecting rod along the x, y, and z directions, J g The Jacobian matrix of the connecting rod force transmission of the parallel mechanism is represented by: Principle 1:

4. The design method of a parallel mechanism with both high normal stiffness and high lateral stiffness according to claim 3, characterized in that: In step S3, by designating the point of action of the external force as the force analysis origin, when the parallel mechanism is acted upon by the lateral force, the force system of the parallel mechanism is expressed as: Among them, ξ r The coefficient representing the radial force, ξ t The coefficient representing the tangential force, the driving force provided by each branch chain is expressed as follows: in, represents the connecting rod torque, Represents the three components of the connecting rod lever arm; the force balance condition is: because, F n > 0, the force balance in the z-axis is expressed as: In order to minimize the connecting rod force, the optimization goal is: Another parallel mechanism is introduced to avoid the motion singularity and withstand the lateral force. The force balance condition of the other parallel mechanism is: in, The sub-table represents the vector components of the hydraulic connecting rod, Express the forces of the three hydraulic connecting rods and solve them by the following equation: Among them, J h is the force transmission Jacobian matrix of the two parallel mechanisms; in order to maximize the lateral stiffness, the design criteria of the other parallel mechanism are as follows:

5. The design method of a parallel mechanism with both high normal stiffness and high lateral stiffness according to claim 4, characterized in that: In step S4, the torque of the six connecting rods and the torque of the hydraulic connecting rod are calculated as follows: in, and is the force torque and The unit spin of and Represents the direction vector of the connecting rod and the hydraulic connecting rod, d l0 Indicates the length of the connecting rod, d h0 and They represent the initial length and extension length of the hydraulic connecting rod respectively; the static equilibrium equation is expressed as follows: Among them, J g is the connecting rod force Jacobian matrix, J h is the hydraulic connecting rod force Jacobian matrix; The driving force vector is expressed as: The parameter combination of the two parallel mechanisms becomes the following optimization problem:

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