An automatic generation method for planar rigid body guidance mechanism based on ground truss model

Abstract

The present invention discloses a method for automatically generating a planar rigid body guide mechanism based on a ground truss model, comprising the following steps: planning the motion trajectory of the end effector, dividing the design domain and initializing the ground truss model; determining the motion form of the driving rod within the model, adding a load to the end effector, and initializing the iterative parameters of the moving asymptote; calculating the strain energy within the model and performing kinematic analysis to calculate the value of the objective function; inputting the objective function and design variables into an optimizer to construct a Lagrangian dual problem and perform analysis and calculation to obtain the updated parameters of the optimized variables; forcibly stopping the iteration process when the number of iteration steps exceeds a set value or the objective function is satisfied; otherwise, continuing to calculate the strain energy within the model and performing kinematic analysis, repeating the iteration until the convergence condition is satisfied; after the iterative design is completed, determining the configuration through motion simulation, achieving mechanism simplification, and completing the design. The present invention reliably implements mechanism selection, dimension generation, and trajectory tracking in the topology optimization process.

Description

Technical Field

[0001] The present invention belongs to the technical field of rigid body guide mechanism design, and in particular relates to a method for automatically generating a planar rigid body guide mechanism based on a ground truss model. Background Art

[0002] A rigid-body guidance mechanism is a mechanical system designed to enable a rigid body to move along a predetermined path. Such mechanisms typically involve kinematic problems in a plane or space, requiring consideration of how to achieve specific motion requirements through the design of linkage mechanisms. Rigid-body guidance mechanisms are crucial in many engineering applications, such as automated machinery, robotics, and robotic arm design.

[0003] The primary design method for rigid-body guided mechanisms is solution domain analysis. This method represents an infinite number of mechanism solutions using a finite solution domain, enabling designers to quickly and accurately select the optimal mechanism that meets design requirements. A preliminary analysis of the mechanism solution domain yields a mechanism property diagram, allowing designers to understand each mechanism's type, minimum transmission angle, rod length ratio, and defect presence. Constraints are then applied to construct a feasible mechanism solution domain, avoiding blind selection of position points. Finally, a performance analysis diagram is drawn based on the feasible mechanism solution domain to determine the optimal solution. However, due to the inextricable kinematic modeling required—the mechanism configuration must be determined in advance—this design approach still requires extensive human research during the preliminary design phase, resulting in low efficiency and high designer skill requirements. It is also not universally applicable or convenient.

[0004] A new approach to the design of rigid body guidance mechanisms is based on topology optimization methods. The most important advantage of this method is that it can simultaneously determine the configuration and size of the mechanism. The idea of ​​using topology optimization to synthesize planar mechanisms was inspired by the design of flexible mechanisms, and during this period, the application of topology optimization witnessed the transition from material structure to mechanism synthesis. On this basis, scholars have focused their research on articulated mechanisms, synthesizing articulated mechanisms based on the basic structure of the truss through optimization technology. However, the previous research objects were all flexible mechanisms and the degrees of freedom were discrete variables. It was difficult to use the Greenbuller equation to achieve topology optimization formed by continuous variables to meet the single degree of freedom condition. Therefore, it is necessary to find a more reliable design method to complete the rapid design of rigid body guidance. Summary of the Invention

[0005] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide a method for automatically generating a planar rigid body guide mechanism based on a ground truss model. The mechanism is represented by elastic rods and zero-length springs, and the automatic generation of the rigid body guide mechanism can be achieved by maximizing the energy transfer efficiency function and restricting the end effector trajectory.

[0006] To achieve the above-mentioned object, the present invention provides a method for automatically generating a planar rigid body guide mechanism based on a ground truss model, comprising:

[0007] Plan the motion trajectory of the end effector and divide the design domain to initialize the ground truss model;

[0008] After determining the motion form of the driving rod in the ground truss model, a load is added to the end effector and the iterative parameters serving as the moving asymptote of the optimization algorithm are initialized;

[0009] Calculating the strain energy in the ground truss model and performing kinematic analysis, calculating the energy transfer efficiency value and weighted trajectory error of each step in the motion process to obtain the objective function value;

[0010] Input the objective function and design variables into the optimizer to construct the Lagrangian dual problem and perform analysis and calculation to obtain the updated parameters of the optimization variables;

[0011] When the number of iteration steps exceeds the set value or the objective function is satisfied, the iteration process is forced to stop and the iterative design ends; otherwise, the strain energy in the model is continued to be calculated and kinematic analysis is performed, and the iteration process is repeated until the convergence condition is met;

[0012] After the iterative design is completed, the configuration is determined through motion simulation to simplify the mechanism and complete the design.

[0013] Preferably, the process of dividing the design domain includes:

[0014] The design domain is discretized using a grid, and each node is connected by a nonlinear rod element. The nodes are connected to the ground through a zero-length spring. The stiffness coefficient of the rod element is determined by the standard cross-sectional area A0 and the coefficient ζ of the rod, and the spring stiffness coefficient is determined by the standard elastic coefficient k0 and the coefficient ξ. The coefficients ζ and ξ range from 0 to 1.

[0015] Preferably, the process of initializing the ground truss model includes:

[0016] The parameters defining the ground truss model include:

[0017] [A1,A2,A3,…,A N ] T =A0ζ p ;

[0018] [k1,k2,k3,…,k M ] T =k0ξ p ;

[0019] ζ=[ζ1,ζ2,ζ3,…,ζ N ] T;

[0020] ξ=[ξ1,ξ2,ξ3,…,ξ M ] T ;

[0021] Where, [A1,A2,A3,…,A N ] T represents the cross-sectional area of ​​all N elastic rods in the model, [k1, k2, k3, ..., k M ] T Represents the stiffness of all M zero-length springs in the model, A0 is the standard cross-sectional area value, k0 is the standard spring stiffness value, parameters ζ and ξ represent the cross-sectional area A and spring stiffness k respectively, and p is the penalty term constant.

[0022] Preferably, adding a load to the end effector includes:

[0023] The load direction is opposite to the motion trajectory speed direction of the end effector.

[0024] Preferably, the load is expressed as:

[0025]

[0026] Where F t ext Represents the vector value of the external load at each step, F0 is the scalar value of the external load, r t represents the actual position of the end effector at step t, represents the specified position of the end effector at step t, t represents the number of simulation steps in the motion analysis, and T represents the total number of simulation steps.

[0027] Preferably, calculating the strain energy within the ground truss model includes calculating the zero-length spring strain energy and the elastic rod strain energy;

[0028] The formulas for calculating the zero-length spring strain energy and the elastic rod strain energy include:

[0029]

[0030]

[0031] Among them E j The calculation method is as follows:

[0032]

[0033] In the formula represents the strain energy of the zero-length spring, represents the strain energy in the elastic rod, X is the displacement of the node relative to the original position, C represents the Young's modulus of the elastic rod, and Ej represents the Green Lagrangian strain of the elastic rod, l j Indicates the current length of the elastic rod, l j,0 Indicates the original length of the elastic rod.

[0034] Preferably, the process of kinematic analysis includes:

[0035] A force balance equation is established, a Jacobian matrix is ​​calculated based on the force balance equation, and the node equilibrium position is solved through Newton iteration.

[0036] Preferably, the process of establishing a force balance equation, calculating a Jacobian matrix according to the force balance equation, and solving the node equilibrium position by Newton iteration includes:

[0037]

[0038]

[0039] v k+1 =v k -F / J

[0040] Where F is the force condition of all nodes, q represents the position information of the node, J is the Jacobian matrix of the force with respect to the node position information, V k+1 is the latest position coordinate in the iteration process, V k is the position coordinate of the previous step in the iteration process.

[0041] Preferably, the calculation expression of the objective function value is:

[0042]

[0043] Where ε is the allowable range of trajectory error, η represents the energy transfer efficiency function, and ψ t represents the trajectory deviation function.

[0044] Preferably, the process of determining the configuration through motion simulation to achieve mechanism simplification includes:

[0045] The mechanism is simplified by checking the internal force of the model at a certain simulation step. If the internal force is less than a preset value, the variable ζ corresponding to the elastic rod is minimized to obtain the final configuration of the mechanism.

[0046] Compared with the prior art, the present invention has the following advantages and technical effects:

[0047] The ground truss model of the present invention is composed of uniformly discretized zero-length springs and nonlinear elastic rods. The springs fix the rod connection nodes on the ground to generate rotary hinges. The design domain is divided according to the target trajectory space, and a reasonable external load is added to the end effector after the drive form is determined. After initializing the iterative parameters of the optimizer, the strain energy is calculated and kinematic analysis is performed. A rigid single-degree-of-freedom mechanism is generated by maximizing the energy transfer efficiency, and the motion trajectory of the end effector is constrained by a weighted trajectory error function. In the MMA optimization algorithm, a Lagrangian dual problem is constructed to transform the constrained optimization problem into a simple unconstrained optimization problem for optimization. The optimization results are post-processed, and the final configuration is obtained by combining the simulation results and the internal forces of the rod. The present invention reliably realizes the mechanism selection, size generation and trajectory tracking of the topology optimization process.

[0048] The present invention uses a topology optimization method to realize the guided synthesis of a rigid body mechanism, solving the problem that traditional topology optimization is limited to the fields of structure and material design and the synthesis of flexible mechanisms.

[0049] The present invention proposes an objective function of energy transfer efficiency. By introducing external force to do work, the maximum rigidity of the mechanism and the single degree of freedom constraint can be achieved only by maximizing the objective function, and normalization is performed, so that the indicators are more universal.

[0050] The present invention adopts a weighted trajectory error function to limit the trajectory of the end effector, and can make the rigid body guide mechanism free to accurately constrain or loosely handle certain positions according to actual conditions.

[0051] The present invention constructs a Lagrangian dual problem in the MMA optimization algorithm to transform the constrained optimization problem into a simple unconstrained optimization problem, thereby improving the convergence speed and avoiding falling into a local optimum. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] The accompanying drawings, which constitute part of this application, are intended to provide a further understanding of this application. The exemplary embodiments and descriptions of this application are intended to explain this application and do not constitute an improper limitation on this application. In the accompanying drawings:

[0053] Figure 1 Schematic diagram of a method flow in an embodiment of the present invention;

[0054] Figure 2 A schematic diagram of a ground truss model according to an embodiment of the present invention;

[0055] Figure 3 This is a schematic diagram of the external force added to the end effector according to an embodiment of the present invention;

[0056] Figure 4 A schematic diagram of strain energy calculation according to an embodiment of the present invention;

[0057] Figure 5A schematic diagram illustrating design conditions for a design mechanism based on the proposed method according to an embodiment of the present invention;

[0058] Figure 6 Schematic diagram of the simplified mechanism and the optimization result of the mechanism designed based on the proposed method according to an embodiment of the present invention. DETAILED DESCRIPTION

[0059] It should be noted that, in the absence of conflict, the embodiments and features of the embodiments in this application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0060] It should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and that, although a logical order is shown in the flowcharts, in some cases, the steps shown or described can be executed in an order different from that shown here.

[0061] like Figure 1-6 As shown, this embodiment provides a method for automatically generating a planar rigid body guide mechanism based on a ground truss model, comprising:

[0062] Plan the motion trajectory of the end effector and divide the design domain to initialize the ground truss model;

[0063] After determining the motion form of the driving rod in the ground truss model, the load is added to the end effector and the iterative parameters that serve as the moving asymptote of the optimization algorithm are initialized;

[0064] Calculate the strain energy in the ground truss model and perform kinematic analysis. Calculate the energy transfer efficiency and weighted trajectory error of each step in the motion process to obtain the objective function value.

[0065] Input the objective function and design variables into the optimizer to construct the Lagrangian dual problem and perform analysis and calculation to obtain the updated parameters of the optimization variables;

[0066] When the number of iteration steps exceeds the set value or the objective function is satisfied, the iteration process is forced to stop and the iterative design ends; otherwise, the strain energy in the model is continued to be calculated and kinematic analysis is performed, and the iteration process is repeated until the convergence condition is met;

[0067] After the iterative design is completed, the configuration is determined through motion simulation to simplify the mechanism and complete the design.

[0068] Furthermore, when planning the motion trajectory of the end effector, in order to realize the trajectory tracking function, a motion trajectory is planned so that the end effector of the mechanism can move along the route.

[0069] like Figure 2 The partitioned design domain shown to initialize the ground truss model includes:

[0070] The design domain is discretized using a grid with a square side length of 9. Each node is connected by a nonlinear rod element, and the node is connected to the ground through a zero-length spring. The stiffness coefficient of the rod element is determined by the standard cross-sectional area A0 of the rod and the coefficient ζ, and the spring stiffness coefficient is determined by the standard elastic coefficient k0 and the coefficient ξ. The coefficients ζ and ξ are set between 0 and 1.

[0071] When defining a representation of a ground truss model, the model is defined by the following parameters:

[0072] [A1,A2,A3,…,A N ] T =A0ζ p

[0073] [k1,k2,k3,…,k M ] T =k0ξ p

[0074] ζ=[ζ1,ζ2,ζ3,…,ζ N ] T

[0075] ξ=[ξ1,ξ2,ξ3,…,ξ M ] T

[0076] Where [A1,A2,A3,…,A N ] T represents the cross-sectional area of ​​all N elastic rods in the model, [k1, k2, k3, ..., k M ] T represents the stiffness of all M zero-length springs in the model, A0 is a standard cross-sectional area value set to 1, k0 is a standard spring stiffness value set to 1000, parameters ζ and ξ are used to represent the cross-sectional area A and stiffness k respectively, and their initial values ​​are both set to 0.5, and p is a penalty term constant of 3;

[0077] Furthermore, the motion form of the driving rod in the model is determined;

[0078] like Figure 3 As shown, the load is added to the end effector in a direction that is always opposite to the velocity of the end effector's motion trajectory;

[0079] The external loads on the end effector are defined as follows:

[0080]

[0081] Where F t ext Represents the vector value of the external load at each step, F0 is the scalar value of the external load and takes 1, r trepresents the actual position of the end effector at step t, represents the specified position of the end effector at step t, t represents the number of simulation steps in the motion analysis, and the total number of simulation steps T is set to 20 in this embodiment.

[0082] Furthermore, the iteration is prepared, and the iteration parameters serving as the moving asymptotes of the optimization algorithm are initialized;

[0083] Calculate strain energy and establish force balance equations, calculate the Jacobian matrix and solve the node equilibrium position through Newton iteration; strain energy includes calculating the strain energy of zero-length springs and elastic rods;

[0084] like Figure 4 The strain energy of the zero-length spring and the elastic rod shown are calculated as follows:

[0085]

[0086]

[0087] Among them E j The calculation method is as follows:

[0088]

[0089] In the formula represents the strain energy of the zero-length spring, represents the strain energy in the elastic rod, X is the displacement of the node relative to the original position, C represents the Young's modulus of the elastic rod, and E j represents the Green Lagrangian strain of the elastic rod, l j Indicates the current length of the elastic rod, l j,0 Indicates the original length of the elastic rod.

[0090] Furthermore, the force balance equation is established and the calculation method for solving the node equilibrium position by solving the Jacobian matrix is ​​as follows:

[0091]

[0092]

[0093] v k+1 =v k -F / J

[0094] Where F is the force condition of all nodes, q represents the position information of the node, J is the Jacobian matrix of the force with respect to the node position information, V k+1 is the latest position coordinate in the iteration process, V k is the position coordinate of the previous step in the iterative process. Through repeated iterations, the force balance equation can be solved to complete the motion simulation analysis.

[0095] Furthermore, the energy transfer efficiency value and weighted trajectory error of each step in the motion process are calculated to obtain the objective function value;

[0096] The objective function value is calculated as follows:

[0097]

[0098] Where ε is the allowable range of trajectory error and is set to 0.1. The energy transfer efficiency function η can be expressed as follows:

[0099]

[0100] In the formula It represents the total work done by the external load when the model moves from the initial position to the current simulation step t*. It represents the total work done by the driving rod when the model moves from the initial position to the current simulation step t*, Represents the total strain energy of the model at the current simulation step t*.

[0101] The trajectory deviation function ψ t It can be expressed as follows:

[0102]

[0103] Where r t represents the actual position of the end effector at step t, represents the specified position of the end effector at step t, and w represents the trajectory error weight coefficient of the end effector in each simulation step.

[0104] Furthermore, the objective function and design variables are input into the optimizer to construct the Lagrangian dual function and perform optimization calculations to obtain the updated parameters of the optimization variables.

[0105] The new Lagrangian dual problem constructed by MMA can be expressed as follows:

[0106]

[0107] Where μ t represents the Lagrange KKT multiplier, and a is the penalty term constant and is set to 1000.

[0108] Furthermore, for convergence judgment, when the number of iteration steps is greater than the set value or the objective function is satisfied, the iteration process is forced to stop and the iterative design ends; otherwise, the iteration process is returned and repeated until the convergence condition is met;

[0109] After the iterative design is completed, the configuration is determined through motion simulation and internal force inspection to achieve mechanism simplification and complete the design. Figure 5-6 shown.

[0110] The calculation method of the model internal force when checking a certain simulation step when simplifying the mechanism is as follows:

[0111]

[0112] S j =CE j

[0113] Where S j is the second Piola-Kirchhoff stress, and C is the Young's modulus of the elastic rod. If the internal force is less than a certain value (here 0.1), the variable ζ corresponding to the elastic rod is minimized to obtain the final configuration of the mechanism.

[0114] In summary, the present invention proposes an efficient topology optimization method for rigid body guide mechanisms, which can complete the design of mechanism configuration and size in one go without relying on engineering experience. The ground truss model consists of uniformly discretized zero-length springs and nonlinear elastic rods. The springs fix the rod connection nodes on the ground to generate rotary hinges; the design domain is divided according to the target trajectory space, and the rods and nodes are uniformly and discretely distributed in the appropriate design domain according to the motion trajectory of the end effector, and the driving mode is determined; an external load is added to the end effector to construct the objective function; the strain energy is calculated and the force balance equation is solved to realize kinematic simulation and calculate the objective function value, while the trajectory error weight coefficient is increased to strengthen the constraints on key positions and weaken the trajectory restrictions in the process; the Lagrangian dual problem is constructed in the MMA optimization algorithm to transform the constrained optimization problem into a simple unconstrained optimization problem and optimize it; the optimization iteration exits the loop after meeting the indicators, and the simulated motion of the optimized planar mechanism is observed to simplify the mechanism and obtain the final configuration. The present invention reliably realizes the mechanism selection, size generation and trajectory tracking of the topology optimization process.

[0115] The above are merely preferred embodiments of the present application, but the scope of protection of the present application is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in this application should be included in the scope of protection of the present application. Therefore, the scope of protection of the present application should be based on the scope of protection of the claims.

Claims

1. A method for automatically generating a planar rigid body guide mechanism based on a ground truss model, characterized in that: include: Plan the motion trajectory of the end effector and divide the design domain to initialize the ground truss model; After determining the motion form of the driving rod in the ground truss model, a load is added to the end effector and the iterative parameters serving as the moving asymptote of the optimization algorithm are initialized; Calculating the strain energy in the ground truss model and performing kinematic analysis, calculating the energy transfer efficiency value and weighted trajectory error of each step in the motion process to obtain the objective function value; Input the objective function and design variables into the optimizer to construct the Lagrangian dual problem and perform analysis and calculation to obtain the updated parameters of the optimization variables; When the number of iteration steps is greater than the set value or the objective function is satisfied, the iteration process is forced to stop and the iterative design ends; Otherwise, continue to calculate the strain energy in the model and perform kinematic analysis, repeating the iterative process until the convergence condition is met; After the iterative design is completed, the configuration is determined through motion simulation to simplify the mechanism and complete the design; The calculation expression of the objective function value is: Where ε is the allowable range of trajectory error, η represents the energy transfer efficiency function, and ψ t represents the trajectory deviation function; parameters ζ and ξ are coefficients ranging from 0 to 1; The energy transfer efficiency function η is expressed as follows: In the formula It represents the total work done by the external load when the model moves from the initial position to the current simulation step t*. It represents the total work done by the driving rod when the model moves from the initial position to the current simulation step t*, Represents the total strain energy of the model at the current simulation step t*; Trajectory deviation function ψ t It is expressed as follows: Where r t represents the actual position of the end effector at step t, represents the prescribed position of the end effector at step t, and w represents the trajectory error weight coefficient of the end effector at each simulation step; Input the objective function and design variables into the optimizer to construct the Lagrange dual function and perform optimization calculation to obtain the updated parameters of the optimization variables; The new Lagrangian dual problem constructed by MMA is expressed as follows: Where μ t represents the Lagrange KKT multiplier, and a is the penalty term constant.

2. The method according to claim 1, characterized in that The process of dividing the design domain and initializing the ground truss model includes: The design domain is discretized using a grid, and each node is connected by a nonlinear rod element, and the node is connected to the ground by a zero-length spring; the stiffness coefficient of the rod element is determined by the standard cross-sectional area A0 of the rod and the coefficient ζ, and the spring stiffness coefficient is determined by the standard spring stiffness value k0 and the coefficient ξ; the parameters ζ and ξ are coefficients ranging from 0 to 1; The parameters defining the ground truss model include: <h2 style=";text-align:left;direction:ltr">[A1,A2,A3,…,A<h2 style=";text-align:left;direction:ltr"> N <h2 style=";text-align:left;direction:ltr"> ]<h2 style=";text-align:left;direction:ltr"> T <h2 style=";text-align:left;direction:ltr"> =A0ζ<h2 style=";text-align:left;direction:ltr"> p <h2 style=";text-align:left;direction:ltr"> ; [k1,k2,k3,…,k M ] T =k0ξ p ; ζ=[ζ1,ζ2,ζ3,…,ζ N ] T ; ξ=[ξ1,ξ2,ξ3,…,ξ M ] T ; Where, [A1,A2,A3,…,A N ] T represents the cross-sectional area of ​​all N elastic rods in the model, [k1, k2, k3, ..., k M ] T represents the stiffness of all M zero-length springs in the model, where A0 is the standard cross-sectional area value, k0 is the standard spring stiffness value, and the parameters ζ and ξ are coefficients ranging from 0 to 1.

3. The method according to claim 1, characterized in that Adding loads to the end effector involves: The load direction is opposite to the motion trajectory speed direction of the end effector.

4. The method according to claim 1, wherein The load is expressed as: Where F t ext Represents the vector value of the external load at each step, F0 is the scalar value of the external load, r t represents the actual position of the end effector at step t, represents the specified position of the end effector at step t, t represents the number of simulation steps in the motion analysis, and T represents the total number of simulation steps.

5. The method according to claim 1, characterized in that Calculating the strain energy within the ground truss model includes calculating the zero-length spring strain energy and the elastic rod strain energy; The formulas for calculating the zero-length spring strain energy and the elastic rod strain energy include: Among them E j The calculation method is as follows: In the formula represents the strain energy of the zero-length spring, represents the strain energy of the elastic rod, X is the offset of the node relative to the original position, C represents the Young's modulus of the elastic rod, E j represents the Green Lagrangian strain of the elastic rod, l j Indicates the current length of the elastic rod, l j,0 Indicates the original length of the elastic rod.

6. The method according to claim 1, wherein The kinematic analysis process includes: A force balance equation is established, a Jacobian matrix is ​​calculated based on the force balance equation, and the node equilibrium position is solved through Newton iteration.

7. The method according to claim 6, characterized in that The process of establishing a force balance equation, calculating a Jacobian matrix based on the force balance equation, and solving a node equilibrium position through Newton iteration includes: v k+1 =v k -F / J Where F is the force condition of all nodes, q represents the position information of the node, J is the Jacobian matrix of the force with respect to the node position information, V k+1 is the latest position coordinate in the iteration process, V k is the position coordinate of the previous step in the iteration process.

8. The method according to claim 1, characterized in that The process of determining the configuration through motion simulation and achieving mechanism simplification includes: The mechanism is simplified by checking the internal force of the model at a certain simulation step. If the internal force is less than a preset value, the variable ζ corresponding to the elastic rod is minimized to obtain the final configuration of the mechanism.

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