Planar rigid body guide mechanism automatic generation method based on ground truss model

Abstract

The invention discloses a ground truss model-based planar rigid body guide mechanism automatic generation method, which comprises the following steps of: planning a motion track of an end effector, and dividing a design domain to initialize a ground truss model; determining a motion form of a driving rod in the model, adding a load to an end effector, and initializing iteration parameters of a moving asymptote; calculating strain energy in the model, performing kinematics analysis, and calculating a target function value; inputting the target function and the design variable into an optimizer to construct a Lagrange duality problem, and analyzing and calculating the Lagrange duality problem to obtain an updated parameter of the optimization variable; when the iteration step number is greater than a set value or the objective function is satisfied, forcibly stopping the iteration process; otherwise, continuously calculating the strain energy in the model and carrying out kinematics analysis, and repeating iteration until the convergence condition is met; and after the iterative design is finished, the configuration is judged through motion simulation, mechanism simplification is realized, and the design is completed. According to the method, mechanism model selection, size generation and trajectory tracking in the topological optimization process are reliably realized.

Description

Technical Field

[0001] The invention belongs to the technical field of rigid body guide mechanism design, and in particular relates to an automatic generation method of a plane rigid body guide mechanism based on a ground truss model. Background Art

[0002] A rigid body guide mechanism is a mechanical system designed to enable a rigid body in the mechanism to move along a predetermined path. This type of mechanism usually involves kinematic problems in a plane or space, and it is necessary to consider how to achieve specific motion requirements through the design of the linkage mechanism. Rigid body guide mechanisms are very important in many engineering applications, such as in automated machinery, robotics, and robot arm design.

[0003] The main design method of rigid body guidance mechanism is solution domain analysis method. This method uses a finite solution domain to represent an infinite number of mechanism solutions, so that designers can quickly and accurately select the optimal mechanism that meets the design requirements. Through the preliminary analysis of the mechanism solution domain, the mechanism attribute diagram is drawn, and the designer can understand the type, minimum transmission angle, rod length ratio, and defects of each mechanism. Then, constraints are imposed to form a feasible mechanism solution domain to avoid the blindness of position point selection. Finally, the mechanism performance analysis diagram is drawn based on the feasible mechanism solution domain to determine the optimal solution of the mechanism. Since it is inseparable from the link of mechanism kinematic modeling, that is, the mechanism configuration must be obtained in advance, this design idea still requires a lot of manpower research work in the preliminary design stage, which is inefficient, has high requirements on the designer's level, and is not universal and convenient.

[0004] A new way to design rigid guidance mechanisms is based on topology optimization. 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 has 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 conditions. Therefore, it is necessary to find a more reliable design method to complete the rapid design of rigid 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 characterized 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 object, the present invention provides a method for automatically generating a planar rigid body guiding 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, adding a load to the end effector and initializing iterative parameters as the moving asymptote of the optimization algorithm;

[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 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 strain energy in the model continues 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 unit, and the node is connected to the ground by a zero-length spring; the stiffness coefficient of the rod unit 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 range of the coefficients ζ and ξ is between 0 and 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] In the formula, [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, It 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] Where 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 Represents 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 according to the force balance equation, and the node equilibrium position is solved by 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, and 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; the strain energy is calculated and kinematic analysis is performed after the iterative parameters of the optimizer are initialized; 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. The Lagrangian dual problem is constructed in the MMA optimization algorithm 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 force of the test 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 rigid body mechanism guided synthesis, solving the problem that traditional topology optimization is limited to the fields of structure and material design and flexible mechanism synthesis.

[0049] The present invention proposes an objective function for 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 guiding mechanism freely and 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 drawings constituting a part of the present application are used to provide a further understanding of the present application. The illustrative embodiments and descriptions of the present application are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0053] Figure 1 A schematic diagram of a method flow chart of 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 for describing design conditions of 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 in the embodiments of the present 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 guiding 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, add the load to the end effector and initialize the iteration parameters as the moving asymptote of the optimization algorithm;

[0064] Calculate the strain energy in the ground truss model and perform kinematic analysis, calculate the energy transfer efficiency value 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 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 strain energy in the model continues 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 partitioning of the design domain to initialize the ground truss model includes:

[0070] The design domain is discretized using a grid. The side length of the square is 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 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 are set to range 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 the initial values ​​are both set to 0.5, and p is the penalty term constant of 3;

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

[0078] like Figure 3 As shown, when adding a load to the end effector, the direction is always opposite to the direction of 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] Further, 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 Jacobian matrix and solve node equilibrium position through Newton iteration; strain energy includes calculation of zero-length spring strain energy and elastic rod strain energy;

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

[0085]

[0086]

[0087] Where 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 Represents 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, and V k+1 is the latest position coordinate in the iteration process, V k It is the position coordinate of the previous step in the iteration 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 represents the total work done by the external load when the model moves from its 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 the design variables are input into the optimizer to construct the Lagrangian dual function and perform optimization calculation to obtain the parameters after the optimization variables are updated;

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

[0106]

[0107] Where μ t represents the Lagrange KKT multiplier, 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 verifying 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), then 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 guiding mechanisms, which can complete the design of mechanism configuration and size at one time without relying on engineering experience. The ground truss model 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 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 the 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 to 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 only preferred specific implementations of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed in the present application should be included in the protection scope of the present application. Therefore, the protection scope of the present application should be based on the protection scope of the claims.

Claims

1. A method for automatically generating a planar rigid body guiding 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, adding a load to the end effector and initializing iterative parameters as the moving asymptote of the optimization algorithm; 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.

2. The method according to claim 1, characterized in that The process of dividing the design domain includes: The design domain is discretized using a grid, and each node is connected by a nonlinear rod unit, and the node is connected to the ground by a zero-length spring; the stiffness coefficient of the rod unit 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 range of the coefficients ζ and ξ is between 0 and 1.

3. The method according to claim 1, characterized in that: The process of initializing the ground truss model includes: 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 ; In the formula, [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.

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

5. The method according to claim 1, characterized in that 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, It 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.

6. The method according to claim 1, characterized in that Calculating the strain energy in 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: Where 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 displacement 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 Represents the original length of the elastic rod.

7. The method according to claim 1, characterized in that The process of kinematic analysis includes: A force balance equation is established, a Jacobian matrix is ​​calculated according to the force balance equation, and the node equilibrium position is solved by Newton iteration.

8. The method according to claim 7, characterized in that The process of establishing a force balance equation, calculating a Jacobian matrix according to the force balance equation, and solving a node equilibrium position by 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, and 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.

9. The method according to claim 1, characterized in that: 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.

10. The method according to claim 1, characterized in that The process of determining the configuration through motion simulation and realizing 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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