Pneumatic bistable coupling motion driver and design method

Abstract

The invention relates to a pneumatic bistable coupling motion driver and a design method, the upper end of an air bag is connected with the middle of a top plate, the lower end of the air bag is connected with the middle of a bottom plate, six limbs are evenly distributed on the outer side of the air bag in the circumferential direction and arranged between the top plate and the bottom plate, and each limb comprises an upper limb part and a lower limb part which are hinged to the hinged fold line position; the upper end of the upper limb part is hinged to the top plate, the lower end of the lower limb part is hinged to the bottom plate, the six limbs are sequenced as L1, L2, L3, L4, L5 and L6 according to the anticlockwise direction, the shapes of the limbs with the sequence numbers as L1 and L4 are completely the same after the limbs with the sequence numbers as L1 and L4 are unfolded, and the whole limbs are of a parallelogram structure; the other limbs are different in shape after being unfolded and are of a structure with the two sides parallel and the upper portion and the lower portion not parallel, and the top plate and the bottom plate are both regular hexagons. According to the invention, twisting motion, telescopic motion and bending motion modes can be coupled, so that the diversity of motion is realized, and the application scene is expanded.

Description

Technical Field

[0001] The invention relates to the field of soft robots, in particular to a pneumatic bistable coupled motion driver and a design method thereof. Background Art

[0002] With the development of robots, safe interactions between humans and robots are becoming more and more frequent. Compared with traditional rigid robots, soft robots have greater degrees of freedom, which makes them more flexible, more safe to interact with, and more adaptable to unstructured environments. The actuator is the core of soft robots, among which pneumatic actuators are easy to obtain and lightweight, and are the earliest driving method used in soft robots. The performance of pneumatic actuators affects the performance parameters and motion diversity of robots.

[0003] With the development of technology, some pneumatic soft actuators based on origami structures have appeared in the prior art. For example, patent CN109129456B discloses a pneumatic bidirectional bending soft actuator based on origami structure, which includes a restriction layer structure and two deformation layer structures with the same structure, and the deformation layer structure is composed of a number of origami structures with the same structure arranged in a straight line. The origami structure can achieve axial expansion and contraction in two different states through folding and the material's own superelasticity. However, the material used in the actuator is still soft materials such as silicone rubber, natural silicone, and rubber.

[0004] Another example is the CN117817648A patent, which discloses a stretch-rotation coupled pneumatic actuator with an origami structure. The actuator is formed by folding a plurality of inflatable modules in an origami manner, each of which includes a ribbon-shaped airbag, and an origami core is arranged inside the ribbon-shaped airbag. The actuator combines linear motion with rotational motion, thus reducing the size of the actuator. However, since there is no actuator with bending motion, this limits the robot's workspace and motion diversity to a certain extent. Summary of the invention

[0005] The purpose of the present invention is to provide a pneumatic bistable coupled motion driver and a design method, which couples torsional motion, telescopic motion, and bending motion modes, thereby achieving motion diversity and expanding application scenarios.

[0006] The objective of the present invention is achieved through the following technical solutions:

[0007] A pneumatic bistable coupled motion driver comprises a top plate, a bottom plate, an airbag and limbs, wherein the upper end of the airbag is connected to the middle of the top plate, and the lower end is connected to the middle of the bottom plate, six limbs are evenly distributed on the outside of the airbag along the circumferential direction and are arranged between the top plate and the bottom plate, the limbs comprise an upper limb portion and a lower limb portion hinged at a hinge fold line position, and the upper end of the upper limb portion is hinged to the top plate, and the lower end of the lower limb portion is hinged to the bottom plate, the six limbs are numbered L1, L2, L3, L4, L5, and L6 in a counterclockwise direction, wherein the limbs numbered L1 and L4 have exactly the same shape after being unfolded and are all parallelogram structures as a whole, and the remaining limbs have different shapes after being unfolded and are all parallel on both sides and non-parallel up and down structures, and the top plate and the bottom plate are both regular hexagons.

[0008] The lower side of the top plate is staggeredly provided with top plate connecting holes and top plate hinge seats along the circumferential direction, wherein the upper end of the upper limb part is hinged to the corresponding top plate hinge seat through a pin shaft.

[0009] The upper side of the bottom plate is staggeredly provided with bottom plate connecting holes and bottom plate hinge seats along the circumferential direction, wherein the lower end of the lower limb part is hinged to the corresponding bottom plate hinge seat through a pin shaft.

[0010] An air nozzle is arranged in the middle of the airbag, and airbag connecting plates are arranged at the upper and lower ends of the airbag.

[0011] An upper hinge groove is arranged at the upper end of the upper limb part of the limb, and a lower hinge groove is arranged at the lower end of the lower limb part, and both side groove walls of the upper hinge groove and the side groove walls of the lower hinge groove are provided with hinge holes for the pin shaft to pass through.

[0012] The upper limb part and the lower limb part both adopt a thick plate origami structure, and the upper limb part and the lower limb part are connected by a film.

[0013] A design method of the pneumatic bistable coupled motion actuator comprises the following steps:

[0014] Step 1: Define the torsion angle θ of the top and bottom plates b , bending angle ζ and height H, where the relative rotation angle between the top plate and the bottom plate when fully folded is the torsion angle θ b , when fully unfolded, the angle between the planes where the top plate and the bottom plate are located is the bending angle ζ, and the distance between the center Ou of the top plate and the center Od of the bottom plate is the height H between the top plate and the bottom plate;

[0015] Step 2: Determine the actual torsion angle θ of the top and bottom plates i Specifically, when fully folded, the top plate corner P corresponding to the limb numbered L1 u,1 and the corresponding bottom plate corner P d,1 Overlap, the top plate corner end P corresponding to the limb number L4 u,4and the corresponding bottom plate corner P d,4 The other corners of the top plate and the other corners of the bottom plate do not overlap, and there is an angle χ i , i = 1, 2...6, assuming that clockwise is positive and counterclockwise is negative, according to the vector angle formula:

[0016]

[0017] Actual torsion angle θ i for:

[0018] θ i =θ b +χ i (2);

[0019] Step 3: Determine the relationship between the bending angle ζ and the parameters of the limb when fully extended, specifically:

[0020] Step 3.1: Establish a coordinate system and determine the positions of the limbs Ai, Bi, Ci, Di, Ei, Di′, Ei′, Fi, Gi and Hi. The coordinate system is established with the center of the bottom plate Od as the origin O and the coordinates are x, y, and z respectively. The equation of the plane where the top plate is located when fully unfolded is:

[0021] -sin0y+cos0z-cos0H=0(3);

[0022] Suppose the corners of the i-th limb after it is fully unfolded are Ai, Bi, Di' and Ei' respectively, and the corners of the i-th limb after it is fully folded are Ai, Bi, Di and Ei respectively, and the two ends of the hinged fold line are Ci and Fi, and a parallel line is drawn from Fi as the starting point, parallel to AiBi and the intersection with BiCi is Hi, and the extension line of DiEi and the extension line of BiAi intersect at the origin of the coordinate O, and the intersection of the extension line of DiEi and AiFi is Gi;

[0023] Step 3.2: At the center of the base plate O d In the coordinate system with the origin O, using homogeneous coordinate transformation, Ai and Bi are expressed as:

[0024] A i =Rot(z,(t-1)×60°+χ i )Trans(x,l d,i )[0 0 0 1] T (4)

[0025] B i =Rot(z,(i-1)×60°+χ i )Trans(x,l d，i +a i )[0 0 0 1]T (5);

[0026] In the above equations (4) and (5), Rot is the rotation homogeneous coordinate transformation, Trans is the translation homogeneous coordinate transformation, and l d，i is the length of OAi, ai is the length of AiBi;

[0027] It is expressed as:

[0028]

[0029] In the above formula (6), αi is the angle between the perpendicular line through Ai and AiFi;

[0030] Another vector / / vector Di′ and Ei′ are and The intersection point with the plane where the top plate is located, so the solution of equations (3) and (6) above is and

[0031] but:

[0032]

[0033] In the above formulas (7) and (8), bi is the length of BiCi, ci is the length of CiDi′, ei is the length of Ei′Fi, and fi is the length of FiAi;

[0034] Step 4: Determine the hinge fold position and torsion angle θ when fully folded i Specifically, for the i-th limb, in the triangle FiHiCi, the sine theorem gives:

[0035] (b i -f i )cos(β i -β i )=a i sin(β i ) (9);

[0036] Similarly, in triangle OAiGi and triangle FiEiGi, we get the following relationship:

[0037]

[0038] In the above equations (9) and (10), λ i For ∠DiEiFi, is ∠EiFiGi.

[0039] In the triangle OGiAi, according to the trigonometric and interior angle theorem:

[0040]

[0041] In the above formula (11), θ i is ∠AiOGi;

[0042] Substituting the above formula (11) into formula (10), we get:

[0043] f i cos(α i -θ i )-l d，i sin(θ i )=e i sin(λ i ) (12);

[0044] Step 5: The simultaneous equations (2), (7), (8), (9) and (12) constitute a set of constraint equations. When the relevant parameters of the i-th limb are given, the size of the limb is determined by solving the above constraint equations through computer software.

[0045] The advantages and positive effects of the present invention are:

[0046] 1. The present invention couples torsional motion, telescopic motion, and bending motion, thereby achieving motion diversity and expanding application scenarios.

[0047] 2. The present invention uses a thick plate origami mechanism as a supporting structure, which can shrink to a smaller volume when air is evacuated. When inflated, the actuator of the present invention is in a fully extended bending state, and the upper and lower limbs of the thick plate origami supporting structure are in a locked state in the same plane, which improves the stability and load-bearing capacity of the actuator to a certain extent.

[0048] 3. The present invention constructs a set of constraint equations through geometric analysis, thereby realizing programmable design of driver dimensions and saving modeling and design time. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 It is a structural schematic diagram of the present invention,

[0050] Figure 2 for Figure 1 Schematic diagram of the top plate structure,

[0051] Figure 3 for Figure 1 Schematic diagram of the bottom plate structure in

[0052] Figure 4 for Figure 1 Schematic diagram of the airbag structure,

[0053] Figure 5 for Figure 1 Schematic diagram of the limb structure in

[0054] Figure 6 for Figure 1 Diagram of the structure of the six limbs in Chinese medicine.

[0055] Figure 7 A schematic diagram of the invention in a folded state when the zero-thickness design is used.

[0056] Figure 8 Schematic diagram of the zero-thickness design of the present invention in an unfolded state,

[0057] Fig. 9 for Figure 7 The projection view of the top plate of the present invention in the folded state on the plane where the bottom plate is located,

[0058] Fig.10 A diagram of the limbs when Ming was designing.

[0059] Among them, 1 is the top plate, 101 is the top plate connecting hole, 102 is the top plate hinge seat, 2 is the bottom plate, 201 is the bottom plate connecting hole, 202 is the bottom plate hinge seat, 3 is the airbag, 301 is the airbag connecting plate, 302 is the air nozzle, 4 is the limb, 401 is the film, 402 is the upper limb part, 403 is the lower limb part, 404 is the hinge hole, 405 is the upper hinge groove, 406 is the lower hinge groove, 407 is the hinge fold line position, and 5 is the pin shaft. DETAILED DESCRIPTION

[0060] The present invention will be further described below in conjunction with the accompanying drawings.

[0061] like Figures 1 to 10 As shown, the present invention includes a top plate 1, a bottom plate 2, an airbag 3 and limbs 4, wherein the upper end of the airbag 3 is connected to the middle of the top plate 1, and the lower end is connected to the middle of the bottom plate 2, six limbs 4 are evenly distributed on the outside of the airbag 3 along the circumferential direction and are arranged between the top plate 1 and the bottom plate 2, and the limbs 4 include an upper limb part 402 and a lower limb part 403 hinged at a hinge fold line position 407, and the upper end of the upper limb part 402 is hinged to the top plate 1, and the lower end of the lower limb part 403 is hinged to the bottom plate 2, as shown in FIG. Figure 6As shown, the six limbs 4 are numbered L1, L2, L3, L4, L5, and L6 in a counterclockwise direction, wherein the limbs 4 numbered L1 and L4 have exactly the same shape after unfolding and are all parallelogram structures as a whole, and the remaining limbs 4 have different shapes after unfolding and are parallel on both sides but not parallel up and down, so that when the present invention is working, the airbag 3 is inflated and deflated to stretch and drive the top plate 1 and the bottom plate 2 to move closer or apart, and at the same time, each limb 4 cooperates to unfold or fold, and when the driver of the present invention is fully extended, it is in a bent state to meet actual needs, and at the same time, during the unfolding or folding process of the above-mentioned limbs 4, the present invention realizes the coupling of torsional motion, telescopic motion, and bending motion, that is, the present invention can realize bending motion, thereby realizing the diversity of motion and expanding the application scenarios. The top plate 1 and the bottom plate 2 are both regular hexagons.

[0062] like Figure 2 As shown, in this embodiment, the lower side of the top plate 1 is staggered with top plate connecting holes 101 and top plate hinge seats 102 along the circumferential direction, wherein the top plate connecting holes 101 are connected to related equipment, and the upper end of the upper limb part 402 is hinged to the corresponding top plate hinge seat 102 through a pin shaft 5.

[0063] like Figure 3 As shown, in this embodiment, the upper side of the base plate 2 is staggered with base plate connection holes 201 and base plate hinge seats 202 along the circumferential direction, wherein the base plate connection 201 is connected to related equipment, and the lower end of the lower limb part 403 is hinged to the corresponding base plate hinge seat 202 through a pin shaft 5.

[0064] like Figure 4 As shown, in this embodiment, the airbag 3 is a Chrislin origami structure, which is a well-known technology in the art. An air nozzle 302 is provided in the middle of the airbag 3 for connecting to a trachea for inflation and deflation. Airbag connecting plates 301 are provided at the upper and lower ends of the airbag 3 and are respectively connected to the top plate 1 and the bottom plate 2.

[0065] like Figure 5 As shown, in this embodiment, an upper hinge groove 405 is provided at the upper end of the upper limb portion 402 of the limb 4, and a lower hinge groove 406 is provided at the lower end of the lower limb portion 403, and the groove walls on both sides of the upper hinge groove 405 and the groove walls on both sides of the lower hinge groove 406 are provided with hinge holes 404 for the pin shaft 5 to pass through.

[0066] In addition, in this embodiment, the upper limb part 402 and the lower limb part 403 both adopt a thick plate origami structure, which are connected by a film 401, and the hinge fold line position 407 forms a film rotating hinge structure.

[0067] The key to the present invention is to drive the limb 4 to fold position to ensure that the actuator meets the relevant design parameters, including bending angle and height. In order to simplify the design, the thickness of the limb 4, the top plate 1 and the bottom plate 2 is ignored. The specific design method of the present invention includes the following steps:

[0068] Step 1: Define the torsion angle θ of the top plate 1 and the bottom plate 2 b , bending angle ζ and height H.

[0069] like Figure 7 As shown, when the present invention is fully folded, the relative rotation angle between the top plate 1 and the bottom plate 2 is the torsion angle θ b ,like Figure 8 As shown, when the present invention is fully unfolded, the angle between the planes where the top plate 1 and the bottom plate 2 are located is the bending angle ζ, and the distance between the center Ou of the top plate 1 and the center Od of the bottom plate 2 is the height H between the top plate 1 and the bottom plate 2 when fully unfolded.

[0070] Step 2: Determine the actual torsion angle θ of the top plate 1 and the bottom plate 2 i .

[0071] In order to ensure that the actuator has a bending effect, the limbs 4 numbered L1 and L4 are designed to be parallelogram structures when unfolded, and the remaining limbs 4 are parallel only on the two sides, but not on the top and bottom. Fig. 9 As shown, when the present invention is fully folded, the corner end P of the top plate 1 corresponding to the limb 4 of serial number L1 u,1 and the corresponding bottom plate 2 corner end P d,1 Overlap, serial number L4 limb 4 corresponding to the top plate 1 corner end P u,4 and the corresponding bottom plate 2 corner end P d,4 However, when the actuator is fully extended, due to the bending angle ζ, Fig. 9 The remaining corners of the top plate 1 and the remaining corners of the bottom plate 2 in the projection view shown do not overlap, and there is an angle χ i (i=2, 3, 5, 6). By establishing a unit length vector coordinate system, assuming that clockwise is positive and counterclockwise is negative, the angle between the top plate 1 corner and the center O can be calculated according to the angle formula between the two vectors. u Connect line r2 and the corner end of the bottom plate 2 to the center O d The angle between the lines r1 is:

[0072]

[0073] The angle of 60° in the above formula (1) is because both the top plate 1 and the bottom plate 2 are regular hexagonal structures, and the six limbs 4 are also evenly distributed along the circumferential direction.

[0074] Therefore, if the subscript i represents different limbs, the actual twist angle θ i We can get:

[0075] θ i =θ b +χ i (2).

[0076] Step 3: When the present invention is in a fully expanded state, determine the parameter relationship between the bending angle ζ and the limb 4.

[0077] Step 3.1: Establish a coordinate system and determine the positions of each point of limb 4, Ai, Bi, Ci, Di, Ei, Di′, Ei′, Fi, Gi and Hi.

[0078] like Figure 7 As shown, a coordinate system is established with the center Od of the base plate 2 as the coordinate origin O.

[0079] like Figure 8 As shown, when the present invention is fully unfolded, the equation of the plane where the top plate 1 is located is:

[0080] -sin(5)y+cos0z-cos(5)H=0(3);

[0081] The above formula (3) is obtained by referring to relevant mathematical formulas and is a well-known technology in the art, where y and z are Figure 7 Coordinates of the coordinate system shown.

[0082] like Fig.10 As shown, assuming that the corner ends of the i-th limb 4 after being fully unfolded are Ai, Bi, Di′ and Ei′ respectively, and the corner ends of the i-th limb 4 after being fully folded are Ai, Bi, Di and Ei respectively, and at the same time, the two ends of the hinged fold line position 407 are Ci and Fi, and a parallel line is drawn with Fi as the starting point, which is parallel to AiBi and intersects with BiCi at Hi. At the same time, the extension line of DiEi and the extension line of BiAi intersect at the coordinate origin O, and the intersection point of the extension line of DiEi and AiFi is Gi. Fig.10 The description is made by taking the i=2 th limb 4 as an example.

[0083] Step 3.2: In the coordinate system with the center Od of the base plate 2 as the origin O, using homogeneous coordinate transformation, Ai and Bi can be expressed as:

[0084] A i =Rot(z,(i-1)×60°+χ i )Trans(x,l d,i )[0 0 0 1] T (4);

[0085] B i =Rot(z,(i-1)×60°+χ i )Trans(x,l d，i +ai )[0 0 0 0 1] T (5);

[0086] In the above equations (4) and (5), Rot is the rotation homogeneous coordinate transformation, Trans is the translation homogeneous coordinate transformation, and l d，i like Fig.10 Shown is the length of OAi, a i like Fig.10 AiBi length shown.

[0087] It can be expressed as:

[0088]

[0089] In the above formula (6), αi is Fig.10 Shown is a perpendicular line through Ai and the angle between the perpendicular line and AiFi.

[0090] Another vector / / vector Di′ and Ei′ are and The intersection point with the plane where the top plate 1 is located, so the solution of equations (3) and (6) above can be obtained and

[0091] but:

[0092]

[0093] In the above formulas (7) and (8), if Fig.10 As shown, bi is the length of BiCi, ci is the length of CiDi′, ei is the length of Ei′Fi, and fi is the length of FiAi.

[0094] Step 4: When the invention is in a fully folded state, determine the relationship between the hinge fold line position 407 and the torsion angle θ.

[0095] Since the limbs 4 of L1 and L4 of the present invention are exactly the same, the other limbs 4 are different, and the definitions of each point are the same, see Fig.10 Limb 4 of L2 is shown. Fig.10 As shown, since A2F2 / / B2C2 (for the i-th limb, i.e., AiFi / / BiCi), F2H2 / / A2B2 (FiHi / / AiBi), we have |F2H2|=a (i.e., |FiHi|=ai), |C2H2|=bf (i.e., |CiHi|=b i -f i), ∠F2H2C2=∠A2B2C2=π / 2-α2 (i.e. ∠FiHiCi=∠AiBiCi=π / 2-α i ).

[0096] For the i-th limb 4, in the triangle FiHiCi, the sine theorem gives:

[0097] (b i -f i )cos(α i -β i )=a i sin(β i ) (9);

[0098] Similarly, the following relationship can be obtained in triangle OAiGi and triangle FiEiGi:

[0099]

[0100] In the above formulas (9) and (10), if Fig.10 As shown, λ i For ∠DiEiFi, is ∠EiFiGi.

[0101] In the triangle OGiAi, according to the trigonometric and interior angle theorem:

[0102]

[0103] In the above formula (11), θ i It is ∠AiOGi.

[0104] Substituting the above formula (11) into formula (10), we get:

[0105] f i cos(α i -θ i )-l d，i sin(θ i )=e i sin(λ i ) (12).

[0106] Step 5: The simultaneous equations (2), (7), (8), (9) and (12) form a set of constraint equations. When ai and α of the i-th limb are given, i ,θ i , ζ and other parameters, the size of the limb 4 can be determined by solving with computer software, and then the limb 4, top plate 1 and bottom plate 2 are processed according to the size, and assembled together with the airbag 3 to form the actuator of the present invention.

Claims

1. A pneumatic bistable coupled motion actuator, characterized in that: The invention comprises a top plate (1), a bottom plate (2), an air bag (3) and limbs (4), wherein the upper end of the air bag (3) is connected to the middle of the top plate (1), and the lower end is connected to the middle of the bottom plate (2); six limbs (4) are evenly distributed on the outside of the air bag (3) along the circumferential direction and are arranged between the top plate (1) and the bottom plate (2); the limbs (4) comprise an upper limb part (402) and a lower limb part (403) hinged at a hinge fold line position (407); and the upper limb part (402) is The end is hinged to the top plate (1), the lower end of the lower limb part (403) is hinged to the bottom plate (2), and the six limbs (4) are numbered L1, L2, L3, L4, L5, and L6 in a counterclockwise direction, wherein the limbs (4) numbered L1 and L4 have exactly the same shape after unfolding and are all parallelogram structures as a whole, and the remaining limbs (4) have different shapes after unfolding and are all parallel on both sides and non-parallel up and down structures, and the top plate (1) and the bottom plate (2) are both regular hexagons.

2. The pneumatic bistable coupled motion actuator according to claim 1, characterized in that: The lower side of the top plate (1) is provided with top plate connection holes (101) and top plate hinge seats (102) staggered along the circumferential direction, wherein the upper end of the upper limb part (402) is hinged to the corresponding top plate hinge seat (102) via a pin shaft (5).

3. The pneumatic bistable coupled motion actuator according to claim 1, characterized in that: The upper side of the bottom plate (2) is provided with bottom plate connection holes (201) and bottom plate hinge seats (202) in an alternating manner along the circumferential direction, wherein the lower end of the lower limb part (403) is hinged to the corresponding bottom plate hinge seat (202) via a pin shaft (5).

4. The pneumatic bistable coupled motion actuator according to claim 1, characterized in that: An air nozzle (302) is provided in the middle of the airbag (3), and airbag connecting plates (301) are provided at the upper and lower ends of the airbag (3).

5. The pneumatic bistable coupled motion actuator according to claim 1, characterized in that: The upper end of the upper limb part (402) of the limb (4) is provided with an upper hinge groove (405), and the lower end of the lower limb part (403) is provided with a lower hinge groove (406), and the groove walls on both sides of the upper hinge groove (405) and the groove walls on both sides of the lower hinge groove (406) are provided with hinge holes (404) for the pin shaft (5) to pass through.

6. The pneumatic bistable coupled motion actuator according to claim 1, characterized in that: The upper limb part (402) and the lower limb part (403) both adopt a thick plate origami structure, and the upper limb part (402) and the lower limb part (403) are connected by a film (401).

7. A method for designing a pneumatic bistable coupled motion actuator according to claim 1, characterized in that: The steps include: Step 1: Define the torsion angle θ of the top plate (1) and the bottom plate (2) b , bending angle ζ and height H, wherein the relative rotation angle between the top plate (1) and the bottom plate (2) when fully folded is the torsion angle θ b When fully unfolded, the angle between the planes of the top plate (1) and the bottom plate (2) is the bending angle ζ, and the center O of the top plate (1) is u and the center of the bottom plate (2) d The distance between them is the height H between the top plate (1) and the bottom plate (2); Step 2: Determine the actual torsion angle θ of the top plate (1) and the bottom plate (2) i Specifically, when fully folded, the corner end P of the top plate (1) corresponding to the limb (4) of sequence number L1 u，1 and the corresponding bottom plate (2) corner end P d，1 Overlap, the limb (4) of number L4 corresponds to the top plate (1) corner end P u，4 and the corresponding bottom plate (2) corner end P d,4 The other corners of the top plate (1) and the other corners of the bottom plate (2) do not overlap, and there is an angle χ i , i = 1, 2...6, assuming that clockwise is positive and counterclockwise is negative, according to the vector angle formula: Actual torsion angle θ i for: i i =θ b +x i (2); Step 3: Determine the bending angle when fully extended; the relationship with the parameters of the limb (4), specifically: Step 3.1: Establish a coordinate system and determine the positions of the limbs (4) at points Ai, Bi, Ci, Di, Ei, Di′, Ei′, Fi, Gi and Hi, with the center of the base plate (2) at O d A coordinate system is established with the origin O and the coordinates x, y, and z respectively, so that the equation of the plane where the top plate (1) lies when fully unfolded is: -sin(ξ)y+cos(ξ)z-cos(ξ)H=0 (3); Assume that the corners of the i-th limb (4) after being fully unfolded are Ai, Bi, Di' and Ei' respectively, and the corners of the i-th limb (4) after being fully folded are Ai, Bi, Di and Ei respectively, and the two ends of the hinge fold line position (407) are Ci and Fi, and a parallel line is drawn from Fi as the starting point, parallel to AiBi and intersecting with BiCi at Hi, and the extension line of DiEi and the extension line of BiAi intersect at the coordinate origin 0, and the extension line of DiEi intersects with AiFi at Gi; Step 3.2: In the coordinate system with the center Od of the base plate (2) as the origin O, use homogeneous coordinate transformation, Ai and Bi are expressed as: AND i =Rot(z,(i-1)×60°+χ i )Trans(x,l d，i )[0 0 0 1] T (4); B i =Rot(z,(i-1)×60°+χ i )Trans(x,l d，i +a i )[0 0 0 1] T (5); In the above equations (4) and (5), Rot is the rotation homogeneous coordinate transformation, Trans is the translation homogeneous coordinate transformation, and l d，i is the length of OAi, ai is the length of AiBi; It is expressed as: In the above formula (6), αi is the angle between the perpendicular line through Ai and AiFi; Another vector / / vector Di′ and Ei′ are and The intersection of the plane of the top plate (1) and the top plate (2) is obtained by combining equations (3) and (6) to obtain and but: In the above formulas (7) and (8), bi is the length of BiCi, ci is the length of CiDi′, ei is the length of Ei′Fi, and fi is the length of FiAi; Step 4: Determine the hinge fold line position (407) and the torsion angle θ when fully folded i Specifically, for the i-th limb (4), in the triangle FiHiCi, the sine theorem gives: (b) i -f i )cos(α i -b i )=a i sin(β i ) (9): Similarly, the following relationship is obtained in triangle OAiGi and triangle FiEiGi: In the above equations (9) and (10), λ i For ∠DiEiFi, is ∠EiFiGi. In the triangle OGiAi, according to the trigonometric and interior angle theorem: In the above formula (11), θ i is ∠AiOGi; Substituting the above formula (11) into formula (10), we get: f i cos(a i -θ i )-l d，i sin(θ i )=e i sin(λ i ) (12); Step 5: The simultaneous equations (2), (7), (8), (9) and (12) constitute a set of constraint equations. When the relevant parameters of the i-th limb are given, the size of the limb (4) is determined by solving the above constraint equations through computer software.

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