CN111262474B - Driving force calculation method of plane hydraulic amplification self-healing electrostatic actuator - Google Patents

Driving force calculation method of plane hydraulic amplification self-healing electrostatic actuator Download PDF

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CN111262474B
CN111262474B CN202010203665.8A CN202010203665A CN111262474B CN 111262474 B CN111262474 B CN 111262474B CN 202010203665 A CN202010203665 A CN 202010203665A CN 111262474 B CN111262474 B CN 111262474B
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CN111262474A (en
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徐静
刘笃信
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Tsinghua University
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Abstract

The invention provides a driving force calculation method of a plane hydraulic amplification self-healing electrostatic actuator, and belongs to the technical field of robot modeling. The method decomposes the driving force into electrostatic force generated by an electrostatic field, elastic force generated by an elastic silica gel outer shell layer and static pressure generated by the elastic silica gel outer shell layer compressing a middle liquid dielectric layer, and calculates each component force respectively and sums to obtain the driving force. The invention considers the influence of various factors such as the size of the actuator, the pre-stretching rate, the thickness of the silica gel shell, the volume of the liquid dielectric medium, related material parameters and the like, and the driving force of the actuator can be calculated by simply measuring the parameters such as the size of the actuator and the like and inputting voltage; the method is simple and efficient, and can quickly calculate the accurate driving force of the planar hydraulic amplification self-healing electrostatic actuator. The invention effectively improves the control precision and the driving efficiency of the actuator applied to the robot system to the robot motion.

Description

Driving force calculation method of plane hydraulic amplification self-healing electrostatic actuator
Technical Field
The invention provides a driving force calculation method of a plane hydraulic amplification self-healing electrostatic actuator, and belongs to the technical field of robot modeling.
Background
The soft actuator has the advantages of light weight, high power density, strong adaptability, high safety and the like, and is widely applied to the fields of man-machine cooperation, man-machine integrated systems, non-structural space operation and the like. Common software actuators include fluid actuators, pneumatic actuators, thermally actuated artificial muscles, dielectric elastomers, and the like. The application of related soft actuators is limited to a certain extent due to the problems of additional pumps required for fluid or pneumatic actuators, low efficiency of thermally actuated artificial muscles, easy breakdown of dielectric elastomers, etc.
A planar hydraulic amplified self-healing electrostatic actuator (HASEL) is a novel soft actuator, and comprises a conductive hydrogel electrode, an elastic silica gel shell and a liquid dielectric medium, wherein the conductive hydrogel electrode is used for applying high-voltage static electricity to the liquid dielectric medium for driving, and the electrostatic force generated by the static electricity and the elastic force of the elastic silica gel shell compress the middle liquid dielectric medium to cause the actuator to deform so as to generate driving force. The structural schematic diagram of the planar hydraulic amplification self-healing electrostatic actuator is shown in fig. 1, and the structural schematic diagram comprises: two layers of conductive hydrogel electrode layers 1 and 6 respectively positioned on the upper surface and the lower surface of the actuator, elastic silica gel outer shells 2 and 5 covered by one layer of electrode respectively connected with each electrode layer, a liquid dielectric layer 4 positioned in the middle of the elastic silica gel outer shells covered by the two layers of electrodes, and elastic silica gel outer shells 3 and 7 positioned on the two sides of the actuator and not covered by the electrodes. Wherein 2, 3, 5 and 7 wrap the liquid dielectric layer in the elastic silica gel outer shell layer. The peak specific work of the actuator is twice that of the traditional dielectric elastomer actuator, and is close to the driving capability of human skeletal muscles. The self-healing type piezoelectric actuator has the advantages of high driving strain, high specific power, self-healing after electrostatic breakdown, low cost and the like, and is expected to be widely applied to systems such as robots and the like.
The soft actuator is used as a power conversion device in the robot system, the driving force output by the soft actuator is directly applied to the robot system body to drive the robot body to move, and the performance of the driving force directly determines the motion capability of the robot system. Therefore, the precise calculation of the driving force of the soft actuator is the premise that the soft actuator is applied to the robot system, so that the control efficiency of the motion of the robot system can be improved, and the structural design of the soft actuator body can be optimized.
As a soft actuator, the driving force of the plane hydraulic amplification self-healing electrostatic actuator is related to various factors such as input voltage, soft materials, external environment and the like. At present, in the aspect of calculating the driving force of a plane hydraulic amplification self-healing electrostatic actuator, the driving force is generally approximate to linear output or a certain section of the selected driving force is approximate to linear. In fact, the driving force of a soft actuator is related to a number of factors, the output of which is non-linear. The driving force calculated through linearization can be applied to some occasions with relaxed requirements, but the error is large, the input-output relationship of the actuator is difficult to obtain accurate correspondence, and the accurate control of the robot system is difficult to realize when the actuator is applied to the robot system for control.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a driving force calculation method of a plane hydraulic amplification self-healing electrostatic actuator. The method is simple and efficient, and can quickly calculate the accurate driving force of the planar hydraulic amplification self-healing electrostatic actuator. The invention effectively improves the control precision and the driving efficiency of the actuator applied to the robot system to the robot motion.
The invention provides a driving force calculation method of a plane hydraulic amplification self-healing electrostatic actuator, which is characterized by comprising the following steps of:
(1) obtaining plane hydraulic amplifying self-healing staticsInitial length L of conductive hydrogel electrode layer of electric actuator1Initial width L of conductive hydrogel electrode layer2Initial thickness L of liquid dielectric layer3(ii) a The initial length and the initial width of the conductive hydrogel electrode layer of the actuator, the elastic silica gel shell layer covered by the electrode and the liquid dielectric layer are equal; obtaining the initial width L 'of the non-electrode covered elastic silica gel outer shell layer'2The volume V of the liquid dielectric layer and the initial thickness D of the elastic silica gel shell layer covered by the single-layer electrode;
pre-stretching the actuator along the width direction of the actuator to obtain the updated width l of the pre-stretched conductive hydrogel electrode layer2The width is fixed by a clamp; after pre-stretching, the widths of the conductive hydrogel electrode layer, the elastic silica gel shell layer covered by the electrode and the liquid dielectric layer after being updated are all l2(ii) a Obtaining the updated thickness of the elastic silica gel shell layer covered by the pre-stretched electrode as d, and the updated width of the elastic silica gel shell layer covered by the non-electrode as l'2；
(2) Calculating the electrostatic force of the plane hydraulic amplification self-healing electrostatic actuator; the method comprises the following specific steps:
(2-1) applying a voltage U to the electrode of the planar hydraulic amplification self-healing electrostatic actuator, wherein the load direction is the length direction of the actuator; after voltage is applied, the updated length of the conductive hydrogel electrode layer is acquired to be l1Obtaining a renewed thickness of the liquid dielectric layer of l3(ii) a The updated lengths of the elastic silica gel outer shell layer and the liquid dielectric layer covered by the electrode are consistent with the updated length of the conductive hydrogel electrode layer, and are l1；
According to the voltage component U of the liquid dielectric layerliqVoltage component U of elastic silica gel outer shell covered by electroderubAnd the relation with the input voltage U, respectively calculating UliqAnd Urub；
Uliq+2Urub＝U
Figure GDA0002796685840000021
Wherein e isliqIs the relative permittivity, e, of the liquid dielectric layerrubIs the relative dielectric constant of the elastic silica gel shell layer;
(2-2) calculation of Electrostatic force F of Electrostatic field in liquid dielectric layeres，liq：
Figure GDA0002796685840000031
Wherein e is0Is a vacuum dielectric constant; gamma rayliq，3Correction factor for electrostatic force of electrostatic field in liquid dielectric layer:
Figure GDA0002796685840000032
(2-3) calculating the Electrostatic force F of the Electrostatic field in the elastic silica gel outer Shell covered with Single-layer electrodees，rub：
Figure GDA0002796685840000033
Wherein, γrub，3The correction coefficient for the electrostatic force of the electrostatic field in the elastic silica gel outer shell covered by the single-layer electrode is as follows:
Figure GDA0002796685840000034
(2-4) according to the results of steps (2-2) and (2-3), the electrostatic force generated in the actuator by the input voltage is equal to the sum of the electrostatic forces of the voltages in the liquid dielectric layer and the elastic silicone shell layer covered by the two layers of electrodes, and the expression is calculated as follows:
Fes＝Fes，liq+2Fes，rub
(3) calculating the elastic force of the plane hydraulic amplification self-healing electrostatic actuator; the method comprises the following specific steps:
(3-1) setting boundary conditions as follows: cauchy principal stress t in thickness direction of actuator 30; cauchy principal stress t of elastic silica gel outer shell covered by electrode in load direction1：
Figure GDA0002796685840000035
Wherein A is an initial parameter of the elastic material related to the shear modulus, JlimIs the maximum average stretch parameter; lambda [ alpha ]1Elongation in the load direction, lambda, of the elastic silicone casing covering the electrodes1＝l1/L1，λ2Elongation in the width direction, lambda, of an elastic silicone casing layer covering the electrodes2＝l2/L2；
(3-2) calculating the elastic force F generated by the elastic silicone rubber outer shell covered by the electrodeel，1：
Figure GDA0002796685840000036
(3-3) calculating the elastic force F generated by the elastic silicone rubber outer shell covered with the non-electrodeel，2：
Figure GDA0002796685840000037
Wherein, t'1The Cauchy principal stress of the elastic silica gel outer shell layer covered by the non-electrode in the load direction;
Figure GDA0002796685840000038
(3-4) calculating the elastic force F generated by the elastic silica gel shell layer according to the results of the steps (3-2) and (3-3)el：
Fel＝Fel，1+Fel，2
(4) Calculating the hydrostatic pressure of the plane hydraulic amplification self-healing electrostatic actuator; the method comprises the following specific steps: :
(4-1) the elastic silicone casing covering the pre-stretched electrodes is approximated by a segment of a circular arc with a chord length l1The length of the arc line segment is consistent with the updating length of the conductive hydrogel electrode layer, the corresponding radius is R, the angle of the arc line segment is theta, and the geometric equation is as follows:
Figure GDA0002796685840000041
Figure GDA0002796685840000042
(4-2) since R > l1Performing second-order Taylor expansion on the two expressions in the step (4-1):
Figure GDA0002796685840000043
Figure GDA0002796685840000044
solving to obtain R and theta;
(4-3) calculation of hydrostatic pressure Fhs：
Figure GDA0002796685840000045
(5) And (4) calculating the driving force of the plane hydraulic amplification self-healing electrostatic actuator by using the results of the steps (2) to (4), wherein the expression is as follows:
F＝Fes+Fel+Fhs。
the invention has the characteristics and beneficial effects that:
the method resolves the driving force into the electrostatic force generated by the electrostatic field, the elastic force generated by the elastic silica gel outer shell layer and the static pressure generated by the elastic silica gel outer shell layer compressing the middle liquid dielectric layer, considers the influence of various factors such as the size, the pre-stretching rate, the thickness of the silica gel outer shell, the volume of the liquid dielectric medium and related material parameters of the actuator, and can calculate the driving force of the actuator by simply measuring the parameters such as the size of the actuator and inputting the voltage. The method has good consistency of the driving force prediction effect and the measured value, and provides a basis for the optimal design and control of the plane hydraulic amplification self-healing electrostatic actuator.
Drawings
Fig. 1 is a structural schematic diagram of a planar hydraulic amplifying self-healing electrostatic actuator.
FIG. 2 is an overall flow diagram of the method of the present invention.
Fig. 3 is a schematic diagram illustrating electrostatic force components of a planar hydraulic amplifying self-healing electrostatic actuator according to an embodiment of the invention.
Fig. 4 is a schematic diagram of the elastic force composition of the planar hydraulic amplification self-healing electrostatic actuator according to the embodiment of the invention.
Fig. 5 is a hydrostatic pressure schematic diagram of a planar hydraulic amplifying self-healing electrostatic actuator according to an embodiment of the invention.
Detailed Description
The invention provides a driving force calculation method of a plane hydraulic amplification self-healing electrostatic actuator, and the invention is further described in detail below by combining specific embodiments.
The invention provides a driving force calculation method of a plane hydraulic amplification self-healing electrostatic actuator, which decomposes the driving force of the actuator into electrostatic force generated by an electrostatic field, elastic force generated by an elastic silica gel outer shell layer and static pressure generated by the elastic silica gel outer shell layer compressing a middle liquid dielectric layer, and is characterized in that the overall flow of the method is shown in figure 2, and the method comprises the following steps:
(1) obtaining the initial length L of the conductive hydrogel electrode layer of the planar hydraulic amplification self-healing electrostatic actuator in the x-axis direction (load direction)1(unit: m), initial width L of conductive hydrogel electrode layer y-axis2(unit: m), initial thickness L of liquid dielectric layer in z-axis direction3(unit: m), schematic view of the direction of xyz axis is shown in FIG. 3, and the initial dimension isThe actuator is manufactured into a size without other operations. The initial length and the initial width of the conductive hydrogel electrode layer, the elastic silica gel shell layer covered by the electrode and the liquid dielectric layer of the actuator are kept consistent. Obtaining an initial width of a non-electrode covered elastic silicone rubber shell layer of L'2(unit: meter), the non-electrode covered elastic silica gel outer shell layer refers to the elastic silica gel outer shell layer distributed on two sides of the electrode (namely the conductive hydrogel electrode layer), and the initial width of the elastic silica gel outer shell layer on each side of the electrode is L'2/2。
And acquiring the volume V (unit: cubic meter) of the liquid dielectric layer of the actuator, and the initial thickness D (unit: meter) of the elastic silica gel shell layer covered by the single-layer electrode on the z axis. Pre-stretching the actuator along the width direction (i.e. y-axis direction) of the actuator (generally, the pre-stretching width is three times of the initial width), and obtaining the y-axis updated width of the pre-stretched conductive hydrogel electrode layer as l2(unit: meter), the width is fixed by a clamp. After pre-stretching, the renewed widths of the conductive hydrogel electrode layer, the elastic silica gel shell layer covered by the electrode and the liquid dielectric layer are kept consistent and are l2. Obtaining the updated thickness of the elastic silica gel shell layer covered by the pre-stretched electrode as d (unit: meter), and the updated width of the elastic silica gel shell layer covered by the non-electrode as l'2(in meters) wherein the renewed width of the elastomeric silicone skin layer on each side of the electrode is l'2/2；
(2) Calculating the electrostatic force of the plane hydraulic amplification self-healing electrostatic actuator, wherein the electrostatic force comprises the electrostatic force of an electrostatic field in a liquid dielectric layer and the electrostatic force of an electrostatic field in an elastic silica gel outer shell layer covered by two layers of electrodes; the method comprises the following specific steps:
and (2-1) applying a voltage U to the electrode of the planar hydraulic amplification self-healing electrostatic actuator, wherein the value range of the voltage U is 0-15 kV. After voltage is applied, the updated length of the conductive hydrogel electrode layer in the x-axis direction is acquired to be l1(unit: meter), obtaining the renewed thickness of the liquid dielectric layer in the z-axis direction as l3(unit: meter). The updated length of the elastic silica gel outer shell layer and the liquid dielectric layer covered by the electrode is consistent with the updated length of the conductive hydrogel electrode layerAre all l1；
The applied voltage acts on the liquid dielectric layer and the elastic silica gel outer shell layer covered by the two layers of electrodes respectively, so that an electrostatic field is abstracted into a planar capacitor structure containing three layers of dielectric substances, and the electrostatic force of the planar hydraulic amplification self-healing electrostatic actuator in the embodiment of the invention is schematically shown in fig. 3. In fig. 3, after the electrostatic field is abstracted to a planar capacitor with three layers of dielectrics, the electrostatic field acts on different dielectric layers to generate different electrostatic forces, including the electrostatic force of the elastic silica gel outer shell covered by two layers of electrodes and the electrostatic force of the liquid dielectric layer. According to the voltage characteristics of parallel plate capacitor, the voltage component U applied to liquid dielectric layer is obtainedliqVoltage component U of elastic silica gel outer shell covered by electroderubRelation to input voltage U:
Uliq+2Urub＝U
Figure GDA0002796685840000061
wherein e isliqIs the relative permittivity, e, of the liquid dielectric layerrubIs the relative dielectric constant of the elastic silicone rubber outer shell layer (the relative dielectric constant is an inherent parameter of the material).
(2-2) calculation of Electrostatic force F of Electrostatic field in liquid dielectric layeres，liq：
Figure GDA0002796685840000062
Wherein e is0Is the dielectric constant in vacuum. Considering the electrostrictive effect, the above formula can be rewritten as:
Figure GDA0002796685840000063
wherein, γliq，3Correction factor for electrostatic force of electrostatic field in liquid dielectric layer:
Figure GDA0002796685840000064
(2-3) calculating the Electrostatic force F of the Electrostatic field in the elastic silica gel outer Shell covered with Single-layer electrodees，rub：
Figure GDA0002796685840000065
Wherein, γrub，3The correction coefficient for the electrostatic force of the electrostatic field in the elastic silica gel outer shell covered by the single-layer electrode is as follows:
Figure GDA0002796685840000066
(2-4) according to the results of steps (2-2) and (2-3), the electrostatic force generated in the actuator by the input voltage is equal to the sum of the electrostatic forces of the voltages in the liquid dielectric layer and the elastic silicone shell layer covered by the two layers of electrodes, and the expression is calculated as follows:
Fes＝Fes，liq+2Fes，rub
according to the working principle of the plane hydraulic amplification self-healing electrostatic actuator, the increase of the electrostatic force causes the reduction of the driving force output, so that the sum of each electrostatic force component and each electrostatic force is a negative value.
(3) Calculating the elastic force of the plane hydraulic amplification self-healing electrostatic actuator; the method comprises the following specific steps:
fig. 4 shows a schematic diagram of the elastic force composition of the planar hydraulic amplification self-healing electrostatic actuator according to the embodiment of the invention. Fig. 4 is a plan view of an elastic silicone rubber outer shell layer of a planar hydraulic amplification self-healing electrostatic actuator, and in fig. 4, after being pre-stretched, two edges of the actuator in the width direction are clamped by clamping plates. The elastic force of the actuator is composed of elastic forces generated by the silica gel elastic bodies in two areas of the elastic silica gel outer shell layer, and the elastic forces are respectively as follows: the elastic force generated by the fully pre-stretched silica gel of the elastic silica gel outer shell covered by the electrode and the elastic force generated by the incompletely pre-stretched silica gel of the elastic silica gel outer shell covered by the non-electrode on the two sides of the electrode. The strain energy generated by the stretching of an elastic material is described by a Gent model, and the elastic force calculation process is as follows:
(3-1) the working process of the pre-stretched elastic silicone rubber casing layer can be simplified to the biaxial stretching problem, and the boundary conditions are set as follows: cauchy principal stress t in thickness direction of actuator 30. Cauchy principal stress t of elastic silica gel outer shell covered by electrode in load direction1：
Figure GDA0002796685840000071
Wherein A is an initial parameter of the elastic material related to the shear modulus, JlimIs the maximum average stretch parameter. The two parameters are measured by experiment to obtain strain data of the silica gel elastic shell material, and then the Gent model is used for measuring material parameters A and JlimAnd (5) performing regression to obtain. Lambda [ alpha ]1Elongation in the load direction (x-axis direction), λ, of the elastic silicone casing layer covering the electrodes1＝l1/L1，λ2Elongation in the y-direction, λ, of an elastic silicone casing covering the electrodes2＝l2/L2。
(3-2) calculating the elastic force F generated by the elastic silicone rubber outer shell covered by the electrodeel，1：
Figure GDA0002796685840000072
(3-3) the elastic force further includes an elastic force F generated by the elastic silicone rubber of the elastic silicone rubber outer shell layer covered by the non-electrodes on both sides of the electrodeel，2The part of the silica gel is not fully pre-stretched, and the initial width of the elastic silica gel shell layers on the two sides of the electrode in the y axis is L'2The width of the elastic silica gel shell silica gel on the two sides of the electrode after pre-stretching operation is l 'on the y axis'2The prestretching rate of the elastic silica gel outer shell layer covered by the non-electrodes on the two sides of the electrode in the y-axis direction
Figure GDA0002796685840000073
Is less than the pre-stretching ratio lambda 'of the elastic silica gel shell layer covered by the electrode in the y-axis direction'2And calculating the elastic force generated by the elastic silica gel outer shell layer covered by the non-electrode:
Figure GDA0002796685840000074
wherein, t'1The Cauchy principal stress of the elastic silica gel outer shell layer covered by the non-electrode in the load direction;
Figure GDA0002796685840000075
(3-4) calculating the elastic force F generated by the elastic silica gel shell layer according to the results of the steps (3-2) and (3-3)el：
Fel＝Fel，1+Fel，2
(4) Calculating the hydrostatic pressure of the plane hydraulic amplification self-healing electrostatic actuator;
the liquid dielectric layer of the actuator is wrapped by the elastic silica gel outer shell layer covered by the two layers of electrodes, the elastic silica gel outer shell layer covered by the electrodes is in an arc-surface state, the elastic force generated by the elastic silica gel outer shell layer of the arc surface can compress the liquid dielectric layer to generate a hydrostatic force, and a hydrostatic force schematic diagram of the plane hydraulic amplification self-healing electrostatic actuator in the embodiment of the invention is shown in fig. 5. The method comprises the following specific steps: :
(4-1) the elastic silica gel shell layer covered by the pre-stretched electrode is approximate to a circular arc segment, and the chord length of the circular arc is l1And the length is consistent with the updated length of the conductive hydrogel electrode layer, the corresponding radius is R, the angle of the arc line segment is theta, and as shown in fig. 5, the geometric equation is as follows:
Figure GDA0002796685840000081
Figure GDA0002796685840000082
(4-2) since R > l1Performing second-order Taylor expansion on the two expressions in the step (4-1):
Figure GDA0002796685840000083
Figure GDA0002796685840000084
and obtaining R and theta after solving.
(4-3) the arc-shaped elastomer wrapping structure generates a hydrostatic force F in the absence of a voltage inpuths：
Figure GDA0002796685840000085
(5) And (4) calculating the driving force of the plane hydraulic amplification self-healing electrostatic actuator by using the results of the steps (2) to (4), wherein the expression is as follows:
F＝Fes+Fel+Fhs
according to the expression, the relation between the driving force output and the input voltage of the plane hydraulic amplification self-healing electrostatic actuator and the displacement of the load direction can be calculated. The method considers the influences of the size, the pre-stretching rate, the thickness of the silica gel shell, the volume of the liquid dielectric medium, relevant material parameters and the like of the actuator, and is a relatively complete plane hydraulic amplification self-healing electrostatic actuator driving force calculation method. The method provides a basis for the application of the plane hydraulic amplification self-healing electrostatic actuator to the optimal design and control of the soft robot.

Claims (1)

1. A driving force calculation method of a plane hydraulic amplification self-healing electrostatic actuator is characterized by comprising the following steps:
(1) obtaining the initial conductive hydrogel electrode layer of a planar hydraulic amplification self-healing electrostatic actuatorLength L1Initial width L of conductive hydrogel electrode layer2Initial thickness L of liquid dielectric layer3(ii) a The initial length and the initial width of the conductive hydrogel electrode layer of the actuator, the elastic silica gel shell layer covered by the electrode and the liquid dielectric layer are equal; obtaining the initial width L 'of the non-electrode covered elastic silica gel outer shell layer'2The volume V of the liquid dielectric layer and the initial thickness D of the elastic silica gel shell layer covered by the single-layer electrode;
pre-stretching the actuator along the width direction of the actuator to obtain the updated width l of the pre-stretched conductive hydrogel electrode layer2The width is fixed by a clamp; after pre-stretching, the widths of the conductive hydrogel electrode layer, the elastic silica gel shell layer covered by the electrode and the liquid dielectric layer after being updated are all l2(ii) a Obtaining the updated thickness of the elastic silica gel shell layer covered by the pre-stretched electrode as d, and the updated width of the elastic silica gel shell layer covered by the non-electrode as l'2；
(2) Calculating the electrostatic force of the plane hydraulic amplification self-healing electrostatic actuator; the method comprises the following specific steps:
(2-1) applying a voltage U to the electrode of the planar hydraulic amplification self-healing electrostatic actuator, wherein the load direction is the length direction of the actuator; after voltage is applied, the updated length of the conductive hydrogel electrode layer is acquired to be l1Obtaining a renewed thickness of the liquid dielectric layer of l3(ii) a The updated lengths of the elastic silica gel outer shell layer and the liquid dielectric layer covered by the electrode are consistent with the updated length of the conductive hydrogel electrode layer, and are l1；
According to the voltage component U of the liquid dielectric layerliqVoltage component U of elastic silica gel outer shell covered by electroderubAnd the relation with the input voltage U, respectively calculating UliqAnd Urub；
Uliq+2Urub＝U
Figure FDA0002796685830000011
Wherein e isliqIs prepared from liquidRelative dielectric constant of bulk dielectric layer, erubIs the relative dielectric constant of the elastic silica gel shell layer;
(2-2) calculation of Electrostatic force F of Electrostatic field in liquid dielectric layeres，liq：
Figure FDA0002796685830000012
Wherein e is0Is a vacuum dielectric constant; gamma rayliq，3Correction factor for electrostatic force of electrostatic field in liquid dielectric layer:
Figure FDA0002796685830000013
(2-3) calculating the Electrostatic force F of the Electrostatic field in the elastic silica gel outer Shell covered with Single-layer electrodees，rub：
Figure FDA0002796685830000021
Wherein, γrub，3The correction coefficient for the electrostatic force of the electrostatic field in the elastic silica gel outer shell covered by the single-layer electrode is as follows:
Figure FDA0002796685830000022
(2-4) according to the results of steps (2-2) and (2-3), the electrostatic force generated in the actuator by the input voltage is equal to the sum of the electrostatic forces of the voltages in the liquid dielectric layer and the elastic silicone shell layer covered by the two layers of electrodes, and the expression is calculated as follows:
Fes＝Fes，liq+2Fes，rub
(3) calculating the elastic force of the plane hydraulic amplification self-healing electrostatic actuator; the method comprises the following specific steps:
(3-1) setting boundary conditions as follows: cauchy of actuator thickness directionPrincipal stress t30; cauchy principal stress t of elastic silica gel outer shell covered by electrode in load direction1：
Figure FDA0002796685830000023
Wherein A is an initial parameter of the elastic material related to the shear modulus, JlimIs the maximum average stretch parameter; lambda [ alpha ]1Elongation in the load direction, lambda, of the elastic silicone casing covering the electrodes1＝l1/L1，λ2Elongation in the width direction, lambda, of an elastic silicone casing layer covering the electrodes2＝l2/L2；
(3-2) calculating the elastic force F generated by the elastic silicone rubber outer shell covered by the electrodeel，1：
Figure FDA0002796685830000024
(3-3) calculating the elastic force F generated by the elastic silicone rubber outer shell covered with the non-electrodeel，2：
Figure FDA0002796685830000025
Wherein, t'1The Cauchy principal stress of the elastic silica gel outer shell layer covered by the non-electrode in the load direction;
Figure FDA0002796685830000026
(3-4) calculating the elastic force F generated by the elastic silica gel shell layer according to the results of the steps (3-2) and (3-3)el：
Fel＝Fel，1+Fel，2
(4) Calculating the hydrostatic pressure of the plane hydraulic amplification self-healing electrostatic actuator; the method comprises the following specific steps: :
(4-1) the elastic silica gel shell layer covered by the pre-stretched electrode is approximate to a circular arc segment, and the chord length of the circular arc is l1The length of the arc line segment is consistent with the updating length of the conductive hydrogel electrode layer, the corresponding radius is R, the angle of the arc line segment is theta, and the geometric equation is as follows:
Figure FDA0002796685830000027
Figure FDA0002796685830000031
(4-2) since R > l1Performing second-order Taylor expansion on the two expressions in the step (4-1):
Figure FDA0002796685830000032
Figure FDA0002796685830000033
solving to obtain R and theta;
(4-3) calculation of hydrostatic pressure Fhs：
Figure FDA0002796685830000034
(5) And (4) calculating the driving force of the plane hydraulic amplification self-healing electrostatic actuator by using the results of the steps (2) to (4), wherein the expression is as follows:
F＝Fes+Fel+Fhs。
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