Dielectric elastomer shape and multi-material soft robot topology optimization method

Abstract

The present invention provides a method for optimizing the shape of a dielectric elastomer and the topology of a multi-material soft robot. The method comprises the following steps: defining a structural design domain of the soft robot, initializing design variables and boundary conditions, penalizing a stiffness matrix using a sequential multi-material interpolation model, calculating the electric field strength of the dielectric elastomer using Ohm's law and a voltage attenuation coefficient model, solving the Maxwell stress field of the dielectric elastomer using a hyperelastic material theoretical model in combination with a thermodynamic framework, mapping the stress field to an equivalent nodal load of the structure by constructing a mechanical equilibrium equation, solving the displacement response of the structure, and establishing a mathematical model for optimizing the driving shape of the dielectric elastomer and the topology of the multi-material soft robot; solving the sensitivity information of the objective function and constraints, and correcting the sensitivity information using a Heaviside filtering method; and updating the design variables using a moving asymptotic algorithm to obtain an optimal body configuration when the topology optimization model meets convergence criteria.

Description

Technical Field

[0001] The present invention relates to the technical field of dielectric elastomer drive mechanism optimization, and in particular to a dielectric elastomer shape and multi-material soft robot topology optimization method. Background Art

[0002] Soft robots generally use intelligent materials such as dielectric elastomers (DE), shape memory alloys, shape memory polymers, etc. as drives. Dielectric elastomers have the advantages of large electro-deformation, high energy density, fast response, and good output controllability, making soft robots driven by them an important research object.

[0003] Dielectric elastomer mechanisms driven by dielectric elastomers are often covered with compliant electrodes to maximize the drive input, while the output is provided by an external mechanism connected to the dielectric elastomer. This mechanism requires a certain strength to support the pre-stretched dielectric elastomer while also being able to deform and output force and displacement. Only by integrating the dielectric elastomer into the mechanism through a rational design can the desired functionality be achieved and its driving effect be maximized.

[0004] Furthermore, as engineering practices place increasing demands on mechanical performance, topology optimization designs based on single-phase materials are no longer sufficient to meet the high-performance demands. Using multiple materials instead of single-phase materials for topology optimization can fully leverage the diverse properties of various materials. By matching multiple materials with varying strengths, engineering mechanical design requirements can be better met. Using topology optimization methods for multi-material mechanical design, the optimal topological configuration for the multi-material mechanism is sought while satisfying constraints, resulting in a multi-material mechanism with optimal performance.

[0005] Currently, the design of soft robots is primarily based on biomimetic principles and the creator's intuition. Designing soft robots with the desired performance requires a deep understanding of material structure and properties, which often makes such designs inefficient and challenging. Therefore, there is an urgent need to develop a theoretical framework suitable for soft robot design. Summary of the Invention

[0006] Based on this, the purpose of the present invention is to provide a dielectric elastomer shape and multi-material soft robot topology optimization method to at least solve the shortcomings of the above-mentioned technology.

[0007] The present invention proposes a dielectric elastomer shape and multi-material soft robot topology optimization method, comprising:

[0008] Step S1: defining the structural design domain, constraint boundary conditions, and initial voltage value of the soft robot, setting the initial values ​​of the design variables, the virtual spring stiffness of the output end, the penalty coefficient, the conductivity coefficient of the electrode area and the non-electrode area, the filter radius, the elastic modulus of the material, the Poisson's ratio, and the volume constraint;

[0009] Step S2: using a multi-material sequence difference model to penalize the elastic modulus of the material unit and calculate the overall stiffness matrix of the body structure of the multi-material soft robot;

[0010] Step S3: Calculate the electric field strength of the dielectric elastomer using Ohm's law and the voltage attenuation coefficient model, solve the Maxwell stress field of the dielectric elastomer using the theoretical model of hyperelastic materials combined with the thermodynamic framework, and calculate the equivalent nodal load of the structure in combination with the equilibrium equation to solve the displacement response of the structure;

[0011] Step S4: constructing a topology optimization model of the dielectric elastomer and the multi-material soft robot with the maximum displacement response as the objective function and the structural volume fraction and strain capacity as constraints;

[0012] Step S5: solving the sensitivity of the objective function with respect to the design variables and the sensitivity of the structural volume and strain energy constraints with respect to the design variables using the topology optimization model of the dielectric elastomer and the multi-material soft robot, and correcting the sensitivity information using a Heaviside filtering method;

[0013] Step S6: Update the design variables using the moving asymptotic algorithm to determine whether the convergence conditions of the optimization algorithm are met:

[0014] If not satisfied, re-execute step S2;

[0015] If satisfied, the topology optimization process ends, and the optimal dielectric elastomer drive shape and multi-material soft robot body configuration are obtained.

[0016] Furthermore, the step S2 includes:

[0017] Normalize the unit density to convert each material parameter into a dimensionless relative value;

[0018] The elastic modulus of material elements is penalized using a multi-material sequence interpolation model;

[0019] Calculate the global stiffness matrix of the multi-material soft robot body structure.

[0020] Furthermore, step S3 includes:

[0021] Determine the conductivity of the electrode area, establish an ordinary differential equation based on the electric field model problem with a voltage attenuation coefficient, and solve the electric field intensity distribution according to the set boundary value conditions;

[0022] Based on the law of conservation of charge, the equilibrium equation is established and solved using the Galerkin method to obtain the state equation of the electric field;

[0023] The constitutive model of the electromechanical coupling system of the dielectric elastomer is derived by combining the hyperelastic material theoretical model with the thermodynamic analysis framework, and the Maxwell stress field of the dielectric elastomer is calculated.

[0024] According to the equilibrium equation, the equivalent node load of the structure is derived, and the displacement field of the structure is obtained.

[0025] Furthermore, the topology optimization models of the dielectric elastomer and the multi-material soft robot are:

[0026] ;

[0027] in, is the objective function, is the position vector of the output displacement, is the displacement vector, T represents the matrix transpose, For the The elastic modulus of the unit, For the The cell density of the unit, is the overall charge density matrix, is the electric field intensity matrix, is the overall stiffness matrix, represents the overall nodal force matrix, is the global transformation matrix, is the Maxwell stress matrix of the dielectric elastomer, is the volume constraint, Indicates the current volume. Indicates the current density, Indicates the permissible volume fraction, is the strain energy constraint, is the current strain energy, is the strain energy defined, Indicates the The cell density of each cell is the density after density filtering, Indicates the The cell density of the unit is Heaviside The density after filtering algorithm processing, Indicates the number of units.

[0028] Furthermore, the step S5 includes:

[0029] The Lagrange multiplier method is used to calculate the sensitivity of the objective function with respect to the design variables, and the sensitivity of the volume constraint and strain energy constraint with respect to the design variables is derived.

[0030] The Heaviside filtering method is used to correct the sensitivity information of the objective function and constraints.

[0031] The present invention proposes a method for optimizing the shape of a dielectric elastomer and the topology of a multi-material soft robot. During the topology optimization process, high-stiffness materials are automatically distributed within the support structure to maintain stability, while low-modulus materials are concentrated in the remaining areas to maximize deformation. Compared to single-material designs, multi-material combinations overcome the contradiction between a single material's stiffness and deformation capacity, achieving increased output displacement within the same volume constraint. This achieves the coordinated optimization of dielectric elastomer drive and structural load-bearing, providing a superior topological configuration design paradigm for high-responsiveness, large-deformation soft robots. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 Flowchart of the dielectric elastomer shape and multi-material soft robot topology optimization method in the first embodiment of the present invention;

[0033] Figure 2 Schematic diagram of the design domain of the dielectric elastomer-driven multi-material soft robot model in the first embodiment of the present invention;

[0034] Figure 3 The dielectric elastomer driving shape and the multi-material soft robot body topology configuration in the first embodiment of the present invention;

[0035] Figure 4 The dielectric elastomer driving shape topology configuration in the first embodiment of the present invention;

[0036] Figure 5 This is the topological configuration of the multi-material soft robot body in the first embodiment of the present invention.

[0037] The following specific embodiments will further illustrate the present invention in conjunction with the above-mentioned drawings. DETAILED DESCRIPTION

[0038] To facilitate understanding of the present invention, the present invention will be described more fully below with reference to the accompanying drawings. The drawings illustrate several embodiments of the present invention. However, the present invention may be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a more thorough and comprehensive understanding of the present invention.

[0039] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one skilled in the art to which this invention pertains. The terms used in this specification of the present invention are for the purpose of describing specific embodiments only and are not intended to limit the present invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

[0040] Example 1

[0041] See also Figure 1 , which shows a dielectric elastomer shape and multi-material soft robot topology optimization method in a first embodiment of the present invention, and the method specifically includes steps S101 to S106:

[0042] S101, defining the structural design domain, constraint boundary conditions, and initial voltage value of the soft robot, setting the initial value of the design variable, the virtual spring stiffness of the output end, the penalty coefficient, the conductivity coefficient of the electrode area and the non-electrode area, the filter radius, the elastic modulus of the material, the Poisson's ratio, and the volume constraint;

[0043] See also Figure 2 , which is a schematic diagram of the design domain of the soft robot in this embodiment.

[0044] S102, using a multi-material sequence difference model to penalize the elastic modulus of the material unit and calculate the overall stiffness matrix of the body structure of the multi-material soft robot;

[0045] In specific implementation, the unit density is normalized to convert each material parameter into a dimensionless relative value:

[0046]

[0047] Where, and Materials The normalized density and the actual density, is the maximum density among all candidate multi-materials, The number of multiple materials.

[0048] The multi-material sequential interpolation model is used to penalize the elastic modulus of the material elements:

[0049]

[0050] Where, and Respectively The elastic modulus and density of each unit, is the penalty coefficient.

[0051] Among them, the proportional coefficient and translation coefficients The calculation formula is:

[0052] ;

[0053] ;

[0054] In the formula, and Materials The elastic modulus and density of the entity, and Respectively indicate materials Normalized elastic modulus and density, .

[0055] Furthermore, the overall stiffness matrix of the multi-material soft robot body structure is calculated.

[0056] S103, using Ohm's law and a voltage attenuation coefficient model to calculate the electric field strength of the dielectric elastomer, using a theoretical model of hyperelastic materials combined with a thermodynamic framework to solve the Maxwell stress field of the dielectric elastomer, and combining the equilibrium equations to calculate the equivalent nodal loads of the structure to solve the displacement response of the structure;

[0057] In specific implementation, the conductivity coefficient of the electrode area is determined, and an ordinary differential equation is established based on the electric field model problem with a voltage attenuation coefficient. The electric field intensity distribution is solved according to the set boundary value conditions. The ordinary differential equation of the electric field model problem with a voltage attenuation coefficient can be expressed as:

[0058] ;

[0059] Where, is the electric potential, is the electric field, is the conductivity matrix, is the voltage attenuation coefficient, is the distance from the power source.

[0060] The boundary conditions are:

[0061] ;

[0062] Where, is the electric potential, is the input voltage.

[0063] Specifically, substitute the boundary value conditions to obtain the electric field intensity distribution:

[0064] ;

[0065] Where, , is the conductivity of the non-electrode part, The distance between input voltage and power supply The ratio of the voltage at .

[0066] Based on the law of conservation of charge, the equilibrium equation is established and solved using the Galerkin method to obtain the state equation of the electric field. According to the law of conservation of charge:

[0067] ;

[0068] Where, and Representing two directions respectively. is the voltage loss, is the voltage attenuation coefficient, , They are 、 The amount of current in the direction, is the conductivity, is the total current change.

[0069] Substituting into Ohm's law, , we can get the Laplace equation for electric potential:

[0070] ;

[0071] In order to avoid directly dealing with the second-order derivative, the Galerkin method is used to transform the differential equation into a weak integral form, and the trial function is selected , convert the differential equation into an integral equation:

[0072] ;

[0073] Among them, each All by It is constructed with the same basis function, and N represents the total number of elements. Discretize into finite elements, each element defines the shape function matrix:

[0074] ;

[0075] in, is the basis function of node i.

[0076] Potential within the cell By node potential Interpolation yields:

[0077] ;

[0078] Where, the node potential , T represents matrix transpose.

[0079] Similarly, we can get the interpolation expression of the trial function:

[0080] ;

[0081] Where, For the The trial function of the unit;

[0082] Solve integral equations using integration by parts and Green's theorem:

[0083] ;

[0084] Where, For the surface The boundary normal of becomes , substituting the weak form into the shape function, we get:

[0085] ;

[0086] Where, the gradient matrix , represents the shape function, represents the external reference potential, is the global transformation matrix. Therefore, the state equation of the electric field can be obtained:

[0087] ;

[0088] in, is the overall charge density matrix, is the electric field intensity matrix, is the overall load vector.

[0089] The constitutive model of the electromechanical coupling system of the dielectric elastomer is derived by combining the hyperelastic material theoretical model with the thermodynamic analysis framework, and the Maxwell stress field of the dielectric elastomer is calculated. The second-order tensor form of the stress of the dielectric elastomer under the action of electromechanical coupling can be expressed as:

[0090] ;

[0091] in, i, j Represents the coordinate direction of the current configuration (after deformation), K are the coordinate directions of the reference configuration (undeformed), Represents the direction in the material coordinate system K Directions mapped to the spatial coordinate system j , Represents the direction in the material coordinate system KDirections mapped to the spatial coordinate system i , The deformation gradient mentioned in is a second-order tensor that represents the mapping relationship before and after material deformation; det is the operator for finding the determinant, which represents the calculation of the deformation gradient F The determinant of is the dielectric constant of vacuum; is the strain energy density function expression of the material, which can be written in the form of deformation gradient; 、 、 The field strength is i, j, k Directional component; is the Kronecker symbol. The entire stress expression is divided into two terms, the first is the elastic force part, the latter is the tensor expression of Maxwell stress.

[0092] According to the equilibrium equation, the equivalent node load of the structure is derived, and the displacement field of the structure is obtained. Among them, the force balance equation of the infinitesimal volume unit is:

[0093] ;

[0094] Where, 、 、 They are 、 、 Directional force. is the stress, in the case of plane Indicates thickness ,and , can be rewritten as .

[0095] In discretization design, Usually, in finite element design, the body forces and traction The resulting external fundamental forces can be expressed as:

[0096] ;

[0097] Where, ,in is the identity matrix. It is a physical array, and traction is not considered in this embodiment. The role of , the above formula can be simplified as:

[0098] ;

[0099] In global form, nodal loads You can use the global transformation matrix and the global stress vector Calculation, that is:

[0100] ;

[0101] Where, the global transformation matrix The unit conversion matrix Assembled obtained.

[0102] S104, constructing a topology optimization model of the dielectric elastomer and the multi-material soft robot with the maximum displacement response as the objective function and the structural volume fraction and strain capacity as constraints;

[0103] In the specific implementation, the maximum output displacement is used as the objective function, and the structural volume fraction and strain energy are used as constraints to establish the dielectric elastomer shape and multi-material soft robot topology optimization design model. The dielectric elastomer shape and multi-material soft robot topology optimization model is:

[0104] ;

[0105] in, is the objective function, is the position vector of the output displacement, is the displacement vector, T represents the matrix transpose, For the The elastic modulus of the unit, For the The cell density of the unit, is the overall charge density matrix, is the electric field intensity matrix, is the overall stiffness matrix, represents the overall nodal force matrix, is the global transformation matrix, is the Maxwell stress matrix of the dielectric elastomer, is the volume constraint, Indicates the current volume. Indicates the current density, Indicates the permissible volume fraction, is the strain energy constraint, is the current strain energy, is the strain energy defined, Indicates the The cell density of each cell is the density after density filtering, Indicates the The cell density of the unit is HeavisideThe density after filtering algorithm processing, Indicates the number of units.

[0106] S105, solving the sensitivity of the objective function with respect to the design variables and the sensitivity of the structural volume and strain energy constraints with respect to the design variables using the topology optimization model of the dielectric elastomer and the multi-material soft robot, and correcting the sensitivity information using a Heaviside filtering method;

[0107] Furthermore, the step S105 specifically includes steps S1051 and S1052:

[0108] S1051, use the Lagrange multiplier method to calculate the sensitivity of the objective function with respect to the design variables, and derive the sensitivity of the volume constraint and strain energy constraint with respect to the design variables;

[0109] S1052, using the Heaviside filtering method to correct the sensitivity information of the objective function and constraints.

[0110] In the specific implementation, the dielectric elastomer shape and multi-material soft robot topology optimization model is used to solve the sensitivity of the objective function with respect to the design variables and the sensitivity of the structural volume and strain energy constraints with respect to the design variables, and the sensitivity information is corrected by the Heaviside filtering method.

[0111] Among them, the sensitivity of the objective function to the design variables is:

[0112] ;

[0113] Where, is the objective function, is the density after density filtering, is the stiffness matrix, is the inverse matrix of the stiffness matrix, is the density after heaviside filtering, is the displacement array, is the global transformation matrix, is the Maxwell stress matrix, is the overall charge density matrix, is the inverse matrix of the overall charge density matrix, is the electric field strength matrix.

[0114] The sensitivity of the volume constraint to the design variables is:

[0115] ;

[0116] Where, is the volume constraint, is the current volume fraction, N is the total number of units;

[0117] The sensitivity of the strain energy constraint to the design variables is:

[0118] ;

[0119] Where, is the strain energy constraint, is the given strain energy;

[0120] S106, using a moving asymptotic algorithm to update the design variables and determine whether the optimization algorithm convergence condition is met:

[0121] If not satisfied, re-execute step S102;

[0122] If satisfied, the topology optimization process ends, and the optimal dielectric elastomer drive shape and multi-material soft robot body configuration are obtained.

[0123] In order to further verify the effectiveness of the dielectric elastomer shape and multi-material soft robot topology optimization method in this embodiment, this embodiment is explained with numerical examples of dielectric elastomer shape and multi-material soft robot topology optimization, and the numerical examples are all two-dimensional structures.

[0124] The semi-symmetric design domain, boundary conditions and input voltage of the dielectric elastomer shape and multi-material soft robot topology optimization model are as follows: Figure 3 As shown, an initial voltage is applied to the left end of the design domain. , in order to facilitate the placement of an object, set a size of vacant areas, L x and L y Representing the design domain x and y The length of the direction, there is also a size of The solid non-design domain is discretized into 160×80 four-node elements, and the minimum filtering radius is , allowable volume fraction , the spring stiffness at the output N / m is used to simulate the stiffness of the clamped object; the parameters of the Neo-Hookean model are set as shear modulus C10 = 16000, incompressibility coefficient D1 = 1 × 1010, the elastic modulus of the material is set to 1 GPa, and the stretching ratio is 3.2 times. The material parameters are shown in Table 1.

[0125] Table 1 Material parameters of the gripper

[0126]

[0127] Figure 3 The dielectric elastomer driving shape and the multi-material soft robot body topology configuration, where the light gray part is the dielectric elastomer driving shape, the black and dark gray parts are the multi-material soft robot body configuration, and its output displacement is 1.8714mm, which is determined by Figure 4 as well as Figure 5 The effectiveness of the optimal dielectric elastomer shape and multi-material soft robot topology optimization obtained in this embodiment was verified.

[0128] In summary, the dielectric elastomer shape and multi-material soft robot topology optimization methods described in the above embodiments of the present invention enable automatic distribution of high-rigidity materials to the support structure to maintain stability, while low-modulus materials are concentrated in the remaining areas to maximize deformation. Compared to single-material designs, multi-material combinations overcome the contradiction between a single material's stiffness and deformation capacity, achieving increased output displacement under the same volume constraint. This achieves the coordinated optimization of dielectric elastomer drive and structural load-bearing, providing a more optimal topological configuration design paradigm for high-responsiveness, large-deformation soft robots.

[0129] The technical features of the above-mentioned embodiments can be combined arbitrarily. In order to make the description concise, not all possible combinations of the technical features in the above-mentioned embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0130] The above-described embodiments merely represent several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that a person skilled in the art could make various modifications and improvements without departing from the spirit of the present application, all of which fall within the scope of protection of the present application. Therefore, the scope of protection of the present patent application shall be determined by the appended claims.

Claims

1. A dielectric elastomer shape and multi-material soft robot topology optimization method, characterized in that: include: Step S1: defining the structural design domain, constraint boundary conditions, and initial voltage value of the soft robot, setting the initial values ​​of the design variables, the virtual spring stiffness of the output end, the penalty coefficient, the conductivity coefficient of the electrode area and the non-electrode area, the filter radius, the elastic modulus of the material, the Poisson's ratio, and the volume constraint; Step S2: using a multi-material sequence difference model to penalize the elastic modulus of the material unit and calculate the overall stiffness matrix of the body structure of the multi-material soft robot; Step S3: Calculate the electric field strength of the dielectric elastomer using Ohm's law and the voltage attenuation coefficient model, solve the Maxwell stress field of the dielectric elastomer using the theoretical model of hyperelastic materials combined with the thermodynamic framework, and calculate the equivalent nodal load of the structure in combination with the equilibrium equation to solve the displacement response of the structure; Step S4: Taking the maximum displacement response as the objective function and the structural volume fraction and strain capacity as constraints, construct a topology optimization model of the dielectric elastomer and the multi-material soft robot, wherein the topology optimization model of the dielectric elastomer and the multi-material soft robot is: ； in, is the objective function, is the position vector of the output displacement, is the displacement vector, T represents the matrix transpose, For the The elastic modulus of the unit, For the The cell density of the unit, is the overall charge density matrix, is the electric field intensity matrix, is the overall stiffness matrix, represents the overall nodal force matrix, is the global transformation matrix, is the Maxwell stress matrix of the dielectric elastomer, is the volume constraint, Indicates the current volume. Indicates the current density, Indicates the permissible volume fraction, is the strain energy constraint, is the current strain energy, is the strain energy defined, Indicates the The cell density of each cell is the density after density filtering, Indicates the The cell density of the unit is Heaviside The density after filtering algorithm processing, Indicates the number of units; Step S5: solving the sensitivity of the objective function with respect to the design variables and the sensitivity of the structural volume and strain energy constraints with respect to the design variables using the topology optimization model of the dielectric elastomer and the multi-material soft robot, and correcting the sensitivity information using a Heaviside filtering method; Step S6: Update the design variables using the moving asymptotic algorithm to determine whether the convergence conditions of the optimization algorithm are met: If not satisfied, re-execute step S2; If satisfied, the topology optimization process ends, and the optimal dielectric elastomer drive shape and multi-material soft robot body configuration are obtained.

2. The dielectric elastomer shape and multi-material soft robot topology optimization method according to claim 1, characterized in that: The step S2 comprises: Normalize the unit density to convert each material parameter into a dimensionless relative value; The elastic modulus of material elements is penalized using a multi-material sequence interpolation model; Calculate the global stiffness matrix of the multi-material soft robot body structure.

3. The dielectric elastomer shape and multi-material soft robot topology optimization method according to claim 1, characterized in that: The step S3 comprises: Determine the conductivity of the electrode area, establish an ordinary differential equation based on the electric field model problem with a voltage attenuation coefficient, and solve the electric field intensity distribution according to the set boundary value conditions; Based on the law of conservation of charge, the equilibrium equation is established and solved using the Galerkin method to obtain the state equation of the electric field; The constitutive model of the electromechanical coupling system of dielectric elastomers is derived by combining the hyperelastic material theoretical model with the thermodynamic analysis framework, and the Maxwell stress field of the dielectric elastomer is calculated. According to the equilibrium equation, the equivalent node load of the structure is derived, and the displacement field of the structure is obtained.

4. The dielectric elastomer shape and multi-material soft robot topology optimization method according to claim 1, characterized in that: The step S5 comprises: The Lagrange multiplier method is used to calculate the sensitivity of the objective function with respect to the design variables, and the sensitivity of the volume constraint and strain energy constraint with respect to the design variables is derived. The Heaviside filtering method is used to correct the sensitivity information of the objective function and constraints.

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