CN110263385B - Mathematical modeling method of soft bidirectional bending pneumatic actuator in bending state - Google Patents

Mathematical modeling method of soft bidirectional bending pneumatic actuator in bending state Download PDF

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CN110263385B
CN110263385B CN201910452745.4A CN201910452745A CN110263385B CN 110263385 B CN110263385 B CN 110263385B CN 201910452745 A CN201910452745 A CN 201910452745A CN 110263385 B CN110263385 B CN 110263385B
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王朝晖
胡家栋
周文
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Zhongshi Micro Technology (Jiaxing) Co.,Ltd.
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Xian Jiaotong University
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Abstract

The invention discloses a digital-analog construction method of a soft bidirectional bending pneumatic actuator in a bending state. Firstly, respectively carrying out mechanical analysis on the expansion side, calculating the tensile deformation and the bending deformation of the expansion side, then carrying out mechanical analysis on the non-expansion side, calculating the compression deformation and the shear deformation of the non-expansion side, and finally establishing the relation between the input pressure and the bending angle of the actuator through integral bending moment conservation. The invention provides a novel mathematical model establishing method suitable for a flexible bidirectional bending pneumatic actuator in a bending state, accurately obtains a relational expression between driving pressure and a bending angle of the bidirectional bending pneumatic actuator, fills the blank of research in the field, and provides a certain theoretical basis for accurate control of the bidirectional bending pneumatic actuator.

Description

Mathematical modeling method of soft bidirectional bending pneumatic actuator in bending state
Technical Field
The invention relates to a soft bidirectional bending pneumatic actuator, in particular to a mathematical modeling method of a soft bidirectional bending pneumatic actuator in a bending state.
Background
The soft bi-directional bending pneumatic actuator is made of a super elastic material, such as silicone rubber. The pneumatic actuator with the soft symmetrical cavity consists of two silica gel modules, wherein the two silica gel modules are provided with a plurality of cavities which are symmetrically distributed and are separated by ribbed plates, and the two silica gel modules are bonded by an ABS plate positioned on a neutral layer. All cavities are connected by a flow channel so that the same pressure is obtained. The change of the pressure of the introduced gas causes the volume change of the cavity, and the soft bidirectional bending pneumatic actuator can realize the self extension or bending. However, the soft pneumatic actuator has the characteristics of large deformation and more degrees of freedom, so that the modeling of the soft pneumatic actuator is difficult. At present, scholars at home and abroad are used for greatly limiting mathematical models of the soft bidirectional bending pneumatic actuator, the accuracy is insufficient, and the control effect of the actuator at the later stage can be influenced.
Therefore, in order to further enhance the control precision of the soft bidirectional bending pneumatic actuator, the establishment of the mathematical model is very important.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a mathematical modeling method of a soft body bidirectional bending pneumatic actuator in a bending state, so that the relation between air pressure and the bending angle of the soft body bidirectional bending pneumatic actuator is accurately determined, and the control precision and the control effect of the soft body bidirectional bending pneumatic actuator in the later period are improved.
The invention is realized by the following technical scheme:
a mathematical modeling method of a soft bidirectional bending pneumatic actuator in a bending state comprises the following steps:
s1, dividing the soft bidirectional bending pneumatic actuator which bends under the action of driving pressure into two parts, wherein one part is an expansion side, and the other part is an unexpanded side;
s2, performing mechanical analysis on the expansion side, and calculating the tensile deformation and the bending deformation of the expansion side to obtain the bending moment of the expansion side;
s3, performing mechanical analysis on the unexpanded side, and calculating the compression deformation and the shear deformation of the unexpanded side to obtain the bending moment of the unexpanded side;
s4, establishing the relation between the driving pressure and the bending angle of the soft bidirectional bending pneumatic actuator through bending moment conservation.
Preferably, the mechanical analysis of the expansion side in S2 specifically includes:
s2.1, performing mechanical analysis on the tensile deformation of the expansion side rib in the radial direction;
and S2.2, performing mechanical analysis on the bending deformation of the expansion side upper wall and the expansion side wall.
Further, S2.1 specifically is:
the radial stress of the expansion side rib is formula (1):
Figure GDA0002527174200000021
wherein the content of the first and second substances,
Figure GDA0002527174200000022
the height of the expanded side cavity, B the width of the cavity, N the number of the expanded side cavities, B the thickness of the ribs, L the original length of the upper wall, t the thickness of the side wall, and p the driving pressure;
obtaining the radial stress and the radial strain of the expansion side rib according to the stress-strain curve of the material used by the actuator
Figure GDA0002527174200000023
The relation between them is formula (2):
Figure GDA0002527174200000024
wherein the content of the first and second substances,
Figure GDA0002527174200000025
the height of the expanded side rib is h, and the original height of the rib is h;
combining the above formula (1) and formula (2) to obtain
Figure GDA0002527174200000026
The value of (c).
Still further, S2.2 specifically is:
the soft bidirectional bending pneumatic actuator bends under the action of driving pressure p, the bending angle is theta, and the deformation delta L of the upper wall of the expansion side is realized according to geometric deformation1Is represented by formula (4):
Figure GDA0002527174200000031
bending strain of expansion side upper wallucIs represented by formula (5):
Figure GDA0002527174200000032
wherein a is the original height of the cavity;
the bending of the upper wall on the expansion side is obtained from the stress-strain curve of the material used for the actuatorStress sigmaucAs in formula (6):
σuc～uc (6)
the bending strain of the expansion side wall is derived by the assumption of constant curvaturescIs represented by the formula (7):
Figure GDA0002527174200000033
wherein y is the distance from the stretching calculation unit layer to the neutral layer;
the bending stress sigma of the expansion side wall can be obtained through the stress-strain curve of the material used by the actuatorscThe relationship to bending strain is (8):
σsc～sc (8)
bending moment M generated by tensile stress of expansion side upper wallucAnd bending moment M generated by bending stress of expansion side wallscFormula (24) and formula (25), respectively:
Figure GDA0002527174200000034
Figure GDA0002527174200000035
still further, the mechanical analysis of the unexpanded side in S3 specifically includes: dividing the deformation of the unexpanded side into two hypothetical steps, the first step being a pure bending deformation, the unexpanded side wall and the unexpanded side upper wall generating a compression pressure; the second step is the free ductile deformation of the unexpanded side wall and the unexpanded side top wall in the circumferential direction, where the compressive energy generated in the first step will be converted into shear energy of the unexpanded side wall.
Still further, in a first step, the compressive strain of the upper wall on the unexpanded sidesIs represented by the formula (10):
Figure GDA0002527174200000041
wherein h is the original height of the rib, s is the free extension distance of the upper wall on the unexpanded side, L is the original length of the upper wall, R is the bending radius of the neutral layer, and when the soft bidirectional bending pneumatic actuator bends under the action of the driving pressure p, the bending angle is theta, and then the bending radius R of the neutral layer is formula (3):
Figure GDA0002527174200000042
obtaining the compressive stress sigma of the upper wall on the unexpanded side from the stress-strain curve of the material used for the actuatorsIs represented by formula (11):
σs～s (11)
when in compression, the tensile calculation unit layer with the distance of y from the upper wall of the unexpanded side is taken as a reference layer, and the length change Delta L of the tensile calculation unit layer is taken asyIs represented by formula (13):
Figure GDA0002527174200000043
when the unexpanded side upper wall free extension distance is s, the length change m of the layer is as shown in equation (14):
Figure GDA0002527174200000044
compressive strain of unexpanded side wallmIs of formula (15):
Figure GDA0002527174200000045
obtaining the compressive stress sigma of the side wall at the non-expansion side according to the stress-strain curve of the material used by the actuatormThe relation with the compressive strain is formula (16):
σm～m (16)
still further, in the second step, during stretching, when s increases from 0 to l,
compression energy E released from the unexpanded side upper wallucRepresented by formula (12):
Figure GDA0002527174200000051
wherein t is the thickness of the side wall, and B is the width of the cavity;
strain energy E released from unexpanded side wallscIs of formula (17):
Figure GDA0002527174200000052
shear strain gamma is generated by the unexpanded side wallxAs in formula (18):
Figure GDA0002527174200000053
wherein x is the distance from the shearing calculation unit layer to the fixed end face;
according to the stress-strain curve of the material used by the actuator, the relation between the shear stress and the shear strain of the side wall at the non-expansion side is obtained as the formula (19):
τx～γx (19)
shear energy E of unexpanded side wallssIs represented by formula (20):
Figure GDA0002527174200000054
according to the conservation of energy, there is the formula (21):
Euc+2Esc＝2Ess (21)
deducing the value of l according to equations (12), (17), (20) and (21);
moment M generated by compressive stress of unexpanded side upper wallsAnd bending moment M generated by compressive stress of unexpanded side wallmFormula (26) and formula (27), respectively:
Figure GDA0002527174200000055
Figure GDA0002527174200000056
Figure GDA0002527174200000061
still further, S4 specifically includes:
when the soft bidirectional bending pneumatic actuator is bent under the action of the driving pressure p and the bending angle is theta, the bending resistance moment M generated by the neutral layer of the soft bidirectional bending pneumatic actuator isnIs represented by formula (22):
Figure GDA0002527174200000062
wherein L is the original length of the upper wall, E is the elastic modulus of the soft material, IzThe force arm of the rectangular section of the neutral layer is shown as the formula (23):
Figure GDA0002527174200000063
wherein B is the width of the cavity, t is the thickness of the side wall, tnIs the thickness of the neutral layer;
bending moment M generated by driving pressure p to neutral layerpIs of formula (28):
Figure GDA0002527174200000064
from the conservation of bending moment, formula (29) is obtained:
Msc+Muc+Ms+Mm+Mn＝Mp (29)
the mathematical relationship between the driving pressure p and the bending angle θ is obtained from equation (29).
Compared with the prior art, the invention has the following beneficial technical effects:
the invention provides a novel mathematical model establishing method suitable for a soft bidirectional bending pneumatic actuator in a bending state, which is used for establishing a mathematical model of the bionic robot fish soft bidirectional bending pneumatic actuator in the bending state by utilizing geometric knowledge and mechanical analysis on the basis of determining the position structure information of the soft bidirectional bending pneumatic actuator, accurately obtaining a relational expression between driving pressure and a bending angle of the bidirectional bending pneumatic actuator and filling the blank of research in the field. The algorithm of the invention is derived from a mathematical equation, and compared with the algorithm of an empirical formula, the method of the invention is tighter. In addition, the invention provides a specific solving method of the method, which is easy to realize in a finite element method. The invention is beneficial to promoting the development of the research on the control method of the soft bidirectional bending pneumatic actuator, thereby providing a certain theoretical basis for the accurate control of the soft bidirectional bending pneumatic actuator, and having important significance for more effectively and accurately controlling the movement of the soft bidirectional bending pneumatic actuator in the research and design process of actual engineering.
Drawings
FIG. 1 is a general block diagram of a flexible bi-directional bending pneumatic actuator;
FIG. 2 is a schematic view of a soft bi-directional bending pneumatic actuator during bending;
FIG. 3 is the stress strain on the expansion side of a soft bi-directional bending pneumatic actuator;
FIG. 4 is the stress strain on the uninflated side of the soft bi-directional bending pneumatic actuator;
FIG. 5 is a graph showing the stress distribution at the end of a soft bi-directional curved pneumatic actuator.
Detailed Description
The present invention will now be described in further detail with reference to specific examples, which are intended to be illustrative, but not limiting, of the invention.
The mathematical modeling method of the soft bi-directional bending pneumatic actuator in the bending state provided in the present example is explained with reference to fig. 1, fig. 2, fig. 3, fig. 4 and fig. 5.
As shown in fig. 1, the soft bidirectional bending pneumatic actuator comprises: upper wall, side wall, neutral layer, rib, cavity and groove.
As shown in FIG. 2, the method divides the soft bi-directional bending pneumatic actuator shown in FIG. 1 into two parts, one part being the inflated side, i.e., the inflated side, and the other part being the uninflated side, i.e., the uninflated side. As shown in fig. 3, first, mechanical analysis is performed on the expansion side, and the tensile deformation and the bending deformation are calculated to obtain the bending moment of the expansion side; then, as shown in fig. 4, the mechanical analysis is performed on the unexpanded side, the compression deformation and the shear deformation of the unexpanded side are calculated, so as to obtain the bending moment of the unexpanded side, and finally, as shown in fig. 5, the relationship between the input pressure and the bending angle of the soft bidirectional bending pneumatic actuator is established through the integral bending moment conservation.
Specifically, the method comprises the following steps:
step 1, stress strain on expansion side at bending
Including tensile deformation of the expansion-side rib in the radial direction and pure bending deformation of the expansion-side upper wall and the expansion-side wall about the vertical axis.
First, the radial stress of the expansion-side rib is formula (1):
Figure GDA0002527174200000081
wherein:
Figure GDA0002527174200000082
the height of the expanded side cavity, B the width of the cavity, N the number of the expanded side cavities, B the thickness of the ribs, L the original length of the upper wall, t the thickness of the side walls, and p the driving pressure.
Then according to the stress-strain curve of the material used for the actuator, the radial stress and the radial strain of the expansion side rib can be obtained
Figure GDA0002527174200000083
The relation between them is formula (2):
Figure GDA0002527174200000084
wherein:
Figure GDA0002527174200000085
the height of the expanded side rib is h, and the original height of the rib is h;
by combining the above formulae (1) and (2), a compound having a structure represented by the formula
Figure GDA00025271742000000810
The value of (c).
The soft bidirectional bending pneumatic actuator bends around a Z axis under the action of driving pressure p, and when the bending angle is theta, the bending radius R of the neutral layer is represented by formula (3):
Figure GDA0002527174200000086
deformation of the upper wall on the expansion side Δ L according to the geometrical deformation1Is represented by formula (4):
Figure GDA0002527174200000087
wherein:
Figure GDA0002527174200000088
is the expanded length of the expanded side upper wall.
Bending strain of expansion side upper wallucIs represented by formula (5):
Figure GDA0002527174200000089
wherein a is the original height of the cavity.
Bending stress σ of expansion-side upper wallucIt can also be read from the stress-strain curve of the material used for the actuator, as shown in equation (6):
σuc～uc (6)
by assuming a constant curvature, the bending strain of the expansion-side wall can be deducedscIs represented by the formula (7):
Figure GDA0002527174200000091
where y is the distance of the tensile calculation unit layer from the neutral layer.
The bending stress sigma of the expansion side wall can be obtained through the stress-strain curve of the material used by the actuatorscThe relationship to bending strain is (8):
σsc～sc (8)
step 2, stress strain of unexpanded side during bending
The deformation of the unexpanded side during bending can be divided into two assumed steps, wherein the first step is pure bending deformation, the unexpanded side wall and the unexpanded side upper wall generate compressive stress, the step is established on an assumed pure bending deformation constraint, the second step is free extension deformation of the unexpanded side wall and the unexpanded side upper wall along the circumferential direction, the unexpanded side is in an unbalanced state after the assumption of the pure bending deformation constraint in the first step is cancelled, and the compressive energy generated in the first step is released and converted into the shearing energy of the unexpanded side wall.
In a first step, the length of the unexpanded upper wall changes by Δ L when it is compressed2Is represented by formula (9):
Figure GDA0002527174200000092
compressive strain of unexpanded side upper wallsIs represented by the formula (10):
Figure GDA0002527174200000093
wherein s is the unexpanded side upper wall free stretch distance;
obtaining the compressive stress sigma of the upper wall on the unexpanded side from the stress-strain curve of the material used for the actuatorsIs represented by formula (11):
σs～s (11)
length change DeltaL of tensile calculation unit layer with distance y from upper wall of non-expansion side in compressionyIs represented by formula (13):
Figure GDA0002527174200000101
when the unexpanded side upper wall free extension distance is s, the length change m of the layer is as shown in equation (14):
Figure GDA0002527174200000102
compressive strain of unexpanded side wallmIs of formula (15):
Figure GDA0002527174200000103
the compressive stress σ of the unexpanded side wall can be obtained from the stress-strain curve of the material used for the actuatormThe relation with the compressive strain is formula (16):
σm～m (16)
in the second step, during the stretching,
when s increases from 0 to l, the compression energy E released by the upper wall on the unexpanded sideucCan be represented by formula (12):
Figure GDA0002527174200000104
strain energy E released from unexpanded side wallscIs of formula (17):
Figure GDA0002527174200000105
when bent, the unexpanded side wall compresses, but when stretchedThe neutral layer remains stationary and the unexpanded side wall will therefore develop a shear strain gammaxAs in formula (18):
Figure GDA0002527174200000106
wherein x is the distance from the shear calculation unit layer to the fixed end face. The fixed end face is the end face of the actuator when the actuator is bent.
According to the stress-strain curve of the material used by the actuator, the relation between the shear stress and the shear strain of the side wall at the non-expansion side can be obtained as shown in the formula (19):
τx～γx (19)
so that when the unexpanded side upper wall is extended by l, the shear energy E of the unexpanded side wallssIs represented by formula (20):
Figure GDA0002527174200000111
according to energy conservation, as in formula (21):
Euc+2ESc＝2ESS (21)
from equations (12), (17), (20) and (21), the value of l can be deduced.
Step 3, conservation of bending moment during bending
When the soft bidirectional bending pneumatic actuator is bent under the action of the driving pressure p and the bending angle is theta, the bending resistance moment M is generated in the neutral layer of the soft bidirectional bending pneumatic actuatornThe bending resistance moment is a bending resistance moment generated by the rigidity of the material, MnIs represented by formula (22):
Figure GDA0002527174200000112
wherein: l is the original length of the upper wall, E is the modulus of elasticity of the soft material, IzThe force arm of the rectangular section of the neutral layer is shown as the formula (23):
Figure GDA0002527174200000113
wherein: t is tnIs the thickness of the neutral layer.
The stress on the expansion side and the non-expansion side of the soft two-way bending pneumatic actuator can generate respective bending moment, and the formula (24) -the formula (27) can be obtained:
Figure GDA0002527174200000114
Figure GDA0002527174200000121
Figure GDA0002527174200000122
Figure GDA0002527174200000123
in the above formula, MucBending moment, M, generated for tensile stress of the expansion-side upper wallscBending moment, M, generated for bending stress of expansion side wallsBending moments, M, produced by compressive stresses on the unexpanded side upper wallmBending moment, M, generated by compressive stress on the unexpanded side wallucAnd MscOnly two variables of applied driving pressure p and bending angle theta are contained, and s at fixed driving pressure p is l, so MsAnd MmOnly one variable of theta is contained.
Bending moment M generated by applied driving pressure p to neutral layerpIs of formula (28):
Figure GDA0002527174200000124
wherein M ispOnly p is a variable, because
Figure GDA0002527174200000125
And also only with p.
Finally, according to the conservation of bending moment, the formula (29) can be obtained:
Msc+Muc+Ms+Mm+Mn＝Mp (29)
the expression (29) only comprises the variables p and theta, so that a mathematical relation between the driving pressure p and the bending angle theta of the soft-body bidirectional bending pneumatic actuator is obtained, namely a mathematical model of the soft-body bidirectional bending pneumatic actuator in the bending state is established.

Claims (6)

1. A mathematical modeling method of a soft bidirectional bending pneumatic actuator in a bending state is characterized by comprising the following steps:
s1, dividing the soft bidirectional bending pneumatic actuator which bends under the action of driving pressure into two parts, wherein one part is an expansion side, and the other part is an unexpanded side;
s2, performing mechanical analysis on the expansion side, and calculating the tensile deformation and the bending deformation of the expansion side to obtain the bending moment of the expansion side;
s3, performing mechanical analysis on the unexpanded side, and calculating the compression deformation and the shear deformation of the unexpanded side to obtain the bending moment of the unexpanded side;
s4, establishing the relation between the driving pressure and the bending angle of the soft bidirectional bending pneumatic actuator through bending moment conservation;
the mechanical analysis of the expansion side in S2 specifically includes:
s2.1, performing mechanical analysis on the tensile deformation of the expansion side rib in the radial direction;
s2.2, performing mechanical analysis on the bending deformation of the expansion side upper wall and the expansion side wall;
s2.1 specifically comprises the following steps:
the radial stress of the expansion side rib is formula (1):
Figure FDA0002527174190000011
wherein the content of the first and second substances,
Figure FDA0002527174190000012
the height of the expanded side cavity, B the width of the cavity, N the number of the expanded side cavities, B the thickness of the ribs, L the original length of the upper wall, t the thickness of the side wall, and p the driving pressure;
obtaining the radial stress and the radial strain of the expansion side rib according to the stress-strain curve of the material used by the actuator
Figure FDA0002527174190000013
The relation between them is formula (2):
Figure FDA0002527174190000014
wherein the content of the first and second substances,
Figure FDA0002527174190000021
the height of the expanded side rib is h, and the original height of the rib is h;
combining the above formula (1) and formula (2) to obtain
Figure FDA0002527174190000022
The value of (c).
2. The mathematical modeling method of a soft bi-directional bending pneumatic actuator in a bending state according to claim 1, wherein S2.2 is specifically:
the soft bidirectional bending pneumatic actuator bends under the action of driving pressure p, the bending angle is theta, and the deformation delta L of the upper wall of the expansion side is realized according to geometric deformation1Is represented by formula (4):
Figure FDA0002527174190000023
bending strain of expansion side upper wallucIs represented by formula (5):
Figure FDA0002527174190000024
wherein a is the original height of the cavity;
obtaining the bending stress sigma of the upper wall of the expansion side according to the stress-strain curve of the material used by the actuatorucAs in formula (6):
σuc～uc (6)
the bending strain of the expansion side wall is derived by the assumption of constant curvaturescIs represented by the formula (7):
Figure FDA0002527174190000025
wherein y is the distance from the stretching calculation unit layer to the neutral layer;
the bending stress sigma of the expansion side wall can be obtained through the stress-strain curve of the material used by the actuatorscThe relationship to bending strain is (8):
σsc～sc (8)
bending moment M generated by tensile stress of expansion side upper wallucAnd bending moment M generated by bending stress of expansion side wallscFormula (24) and formula (25), respectively:
Figure FDA0002527174190000026
Figure FDA0002527174190000031
3. the method of claim 2, wherein the step of performing a mechanical analysis on the unexpanded side at S3 comprises: dividing the deformation of the unexpanded side into two hypothetical steps, the first step being a pure bending deformation, the unexpanded side wall and the unexpanded side upper wall generating a compression pressure; the second step is the free ductile deformation of the unexpanded side wall and the unexpanded side top wall in the circumferential direction, where the compressive energy generated in the first step will be converted into shear energy of the unexpanded side wall.
4. The mathematical modeling method of a soft bi-directional bending pneumatic actuator in a bending state according to claim 3,
in the first step, compressive strain of the upper wall on the unexpanded sidesIs represented by the formula (10):
Figure FDA0002527174190000032
wherein h is the original height of the rib, s is the free extension distance of the upper wall on the unexpanded side, L is the original length of the upper wall, R is the bending radius of the neutral layer, and when the soft bidirectional bending pneumatic actuator bends under the action of the driving pressure p, the bending angle is theta, and then the bending radius R of the neutral layer is formula (3):
Figure FDA0002527174190000033
obtaining the compressive stress sigma of the upper wall on the unexpanded side from the stress-strain curve of the material used for the actuatorsIs represented by formula (11):
σs～s (11)
when in compression, the tensile calculation unit layer with the distance of y from the upper wall of the unexpanded side is taken as a reference layer, and the length change Delta L of the tensile calculation unit layer is taken asyIs represented by formula (13):
Figure FDA0002527174190000034
when the unexpanded side upper wall free extension distance is s, the length change m of the layer is as shown in equation (14):
Figure FDA0002527174190000041
compressive strain of unexpanded side wallmIs of formula (15):
Figure FDA0002527174190000042
obtaining the compressive stress sigma of the side wall at the non-expansion side according to the stress-strain curve of the material used by the actuatormThe relation with the compressive strain is formula (16):
σm～m (16)。
5. the method of claim 4, wherein in the second step, when s is increased from 0 to l during stretching,
compression energy E released from the unexpanded side upper wallucRepresented by formula (12):
Figure FDA0002527174190000043
wherein t is the thickness of the side wall, and B is the width of the cavity;
strain energy E released from unexpanded side wallscIs of formula (17):
Figure FDA0002527174190000044
shear strain gamma is generated by the unexpanded side wallxAs in formula (18):
Figure FDA0002527174190000045
wherein x is the distance from the shearing calculation unit layer to the fixed end face;
according to the stress-strain curve of the material used by the actuator, the relation between the shear stress and the shear strain of the side wall at the non-expansion side is obtained as the formula (19):
τx～γx (19)
shear energy E of unexpanded side wallssIs represented by formula (20):
Figure FDA0002527174190000051
according to the conservation of energy, there is the formula (21):
Euc+2ESc＝2ESS (21)
deducing the value of l according to equations (12), (17), (20) and (21);
moment M generated by compressive stress of unexpanded side upper wallsAnd bending moment M generated by compressive stress of unexpanded side wallmFormula (26) and formula (27), respectively:
Figure FDA0002527174190000052
Figure FDA0002527174190000053
6. the method of claim 5, wherein S4 specifically comprises:
when the soft bidirectional bending pneumatic actuator is bent under the action of the driving pressure p and the bending angle is theta, the bending resistance moment M generated by the neutral layer of the soft bidirectional bending pneumatic actuator isnIs represented by formula (22):
Figure FDA0002527174190000054
wherein L is the original length of the upper wall, E is the elastic modulus of the soft material, IzThe force arm of the rectangular section of the neutral layer is shown as the formula (23):
Figure FDA0002527174190000055
wherein B is the width of the cavity, t is the thickness of the side wall, tnIs the thickness of the neutral layer;
bending moment M generated by driving pressure p to neutral layerpIs of formula (28):
Figure FDA0002527174190000056
from the conservation of bending moment, formula (29) is obtained:
Msc+Muc+Ms+Mm+Mn＝Mp (29)
the mathematical relationship between the driving pressure p and the bending angle θ is obtained from equation (29).
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