Pneumatic bistable coupled motion actuator and design method

Abstract

The present invention relates to a pneumatic bistable coupled motion driver and design method, wherein the upper end of an airbag is connected to the middle of a top plate, and the lower end is connected to the middle of a bottom plate, and six limbs are evenly distributed outside the airbag along the circumferential direction and arranged between the top plate and the bottom plate, the limbs including 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, and the six limbs are numbered L1, L2, L3, L4, L5, and L6 in a counterclockwise order, wherein the limbs numbered L1 and L4 have exactly the same shape after unfolding and are all parallelogram structures as a whole, and the remaining limbs have different shapes after unfolding and are all parallel on both sides but non-parallel up and down, and the top plate and bottom plate are both regular hexagons. The present invention can couple torsional motion, telescopic motion, and bending motion modes, thereby achieving motion diversity and expanding application scenarios.

Description

Technical Field

[0001] This invention relates to the field of soft robots, specifically to a pneumatic bistable coupled motion actuator and its design method. Background Technology

[0002] With the development of robotics, safe interactions between humans and robots are becoming increasingly frequent. Compared to traditional rigid robots, soft robots possess greater degrees of freedom, resulting in greater flexibility, higher interaction safety, and stronger adaptability to unstructured environments. Actuators are the core of soft robots, with pneumatic actuators being readily available and lightweight, making them the earliest actuation method used in soft robots. The performance of pneumatic actuators directly impacts the robot's performance parameters and motion diversity.

[0003] With technological advancements, some pneumatic soft actuators based on origami structures have emerged in existing technologies. For example, patent CN109129456B discloses a pneumatic bidirectional bending soft actuator based on an origami structure, which includes a constraint layer structure and two identical deformation layer structures. The deformation layer structure is composed of several linearly arranged identical origami structures. The origami structure can achieve axial extension and contraction in two different states through folding and unfolding, as well as the material's own hyperelasticity. However, the actuator still uses soft materials such as silicone rubber, natural silicone, and rubber.

[0004] For example, patent CN117817648A discloses a stretch-rotation coupling pneumatic actuator with an origami structure. This actuator is composed of multiple inflatable modules folded using an origami method. Each inflatable module includes a strip-shaped airbag with an origami core material inside. This actuator combines linear and rotational motion, reducing its size. However, the lack of an actuator with bending motion somewhat limits the robot's workspace and the diversity of its movements. Summary of the Invention

[0005] The purpose of this invention is to provide a pneumatic bistable coupled motion actuator and its design method, which couples torsional motion, extensional motion and bending motion, thereby achieving motion diversity and expanding application scenarios.

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

[0007] A pneumatic bistable coupled motion actuator includes a top plate, a bottom plate, an airbag, and limbs. 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 around the outside of the airbag along the circumference and are located between the top plate and the bottom plate. Each limb includes an upper limb and a lower limb hinged at the hinge line. The upper end of the upper limb is hinged to the top plate, and the lower end of the lower limb is hinged to the bottom plate. The six limbs are numbered L1, L2, L3, L4, L5, and L6 in a counterclockwise direction. The limbs numbered L1 and L4 have the same shape after unfolding and are both parallelogram structures. The other limbs have different shapes after unfolding and are all parallel on both sides but not parallel on top and bottom. The top plate and the bottom plate are both regular hexagons.

[0008] The lower side of the top plate is provided with top plate connecting holes and top plate hinge seats in an alternating circumferential direction, wherein the upper end of the upper body is hinged to the corresponding top plate hinge seat by a pin.

[0009] The upper side of the base plate is provided with base plate connecting holes and base plate hinge seats in an alternating circumferential direction, wherein the lower end of the lower limb is hinged to the corresponding base plate hinge seat by a pin.

[0010] The airbag has an air nozzle in the middle and airbag connecting plates at both the upper and lower ends.

[0011] The upper part of the limb is provided with an upper hinge groove at the upper end and the lower part of the limb is provided with a lower hinge groove at the lower end. The groove walls on both sides of the upper hinge groove and the groove walls on both sides of the lower hinge groove are provided with hinge holes for the pin shaft to pass through.

[0012] Both the upper and lower limbs adopt a thick-plate origami structure, and the upper and lower limbs are connected by a thin film.

[0013] A design method for the aforementioned pneumatic bistable coupled motion actuator includes the following steps:

[0014] Step 1: Define the torsion angle θ between the top and bottom plates. b The bending angle ζ and height H, where the relative rotation angle between the top and bottom plates when fully folded is the torsion angle θ. b When fully unfolded, the angle between the planes containing the top plate and the bottom plate 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 limb number L1 u,1 and the corresponding corner P of the base plate d,1 Overlapping, the limb with serial number L4 corresponds to the top plate corner P u,4and the corresponding corner P of the base plate d,4 The remaining corners of the top plate and the remaining corners of the bottom plate do not coincide, and there is an included angle χ. i Let i = 1, 2, ..., 6. Assuming clockwise is positive and counterclockwise is negative, according to the formula for the angle between vectors:

[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 when fully extended and the parameters of the limb, specifically:

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

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

[0022] Let Ai, Bi, Di′, and Ei′ be the corners of the i-th limb when it is fully extended, and let Ai, Bi, Di, and Ei be the corners of the i-th limb when it is fully folded. Let Ci and Fi be the two ends of the hinge line. Let Hi be the point where a line parallel to AiBi and BiCi is drawn with Fi as the starting point. Let the extension of DiEi and the extension of BiAi intersect at the origin O, and let Gi be the point where the extension of DiEi intersects AiFi.

[0023] Step 3.2: At the center O of the base plate d In a coordinate system with origin O, using homogeneous coordinate transformation, Ai and Bi are represented 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 equations (4) and (5) above, Rot represents a rotational homogeneous coordinate transformation, Trans represents a translational homogeneous coordinate transformation, and l d，i Let OAi be the length of Ai, and ai be the length of AiBi.

[0027] Represented as:

[0028]

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

[0030] Other vectors / / vector Di′ and Ei′ are respectively and The intersection point with the plane containing the top plate, therefore, by simultaneously solving equations (3) and (6), we can obtain the solution. and

[0031] but:

[0032]

[0033] In equations (7) and (8) above, 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 position of the hinge fold line and the torsion angle θ when fully folded. i The relationship is as follows: For the i-th limb, in triangle FiHiCi, by the Law of Sines:

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

[0036] Similarly, the following relationship is obtained in triangles OAiGi and FiEiGi:

[0037]

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

[0039] In triangle OGiAi, by the trigonometric sum of interior angles theorem, we have:

[0040]

[0041] In equation (11) above, θ i For ∠AiOGi;

[0042] Substituting equation (11) into equation (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: Solve equations (2), (7), (8), (9) and (12) to form 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 using computer software.

[0045] The advantages and positive effects of this invention are as follows:

[0046] 1. This invention couples torsional motion, extension motion, and bending motion, thereby achieving diversity of motion and expanding application scenarios.

[0047] 2. This invention uses a thick-plate origami mechanism as a support structure, which can shrink to obtain a smaller volume when deflated. When inflated, the actuator of this invention is in a fully extended bent state, and the upper and lower limbs of the thick-plate origami support structure are in a locked state on the same plane, which improves the stability and load-bearing capacity of the actuator to a certain extent.

[0048] 3. This invention constructs a set of constraint equations through geometric analysis, enabling programmable design of the actuator dimensions and saving modeling and design time. Attached Figure Description

[0049] Figure 1 This is a schematic diagram of the structure of the present invention.

[0050] Figure 2 for Figure 1 A schematic diagram of the top slab structure.

[0051] Figure 3 for Figure 1 A schematic diagram of the base plate structure.

[0052] Figure 4 for Figure 1 A schematic diagram of the airbag structure.

[0053] Figure 5 for Figure 1 A diagram of the limb structure in the image.

[0054] Figure 6 for Figure 1 A schematic diagram of the limb structure of a Form Six student.

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

[0056] Figure 8 A schematic diagram of the invention in its unfolded state during zero-thickness design.

[0057] Figure 9 for Figure 7 The projection view of the top plate onto the plane of the bottom plate when the present invention is in a folded state.

[0058] Figure 10 A diagram of Ming's limbs during the design process.

[0059] Wherein, 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 membrane, 402 is the upper limb, 403 is the lower limb, 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. Detailed Implementation

[0060] The invention will now be described in further detail with reference to the accompanying drawings.

[0061] like Figures 1-10 As shown, the present invention includes a top plate 1, a bottom plate 2, an airbag 3, and limbs 4. 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 along the circumference outside the airbag 3 and positioned between the top plate 1 and the bottom plate 2. Each limb 4 includes an upper limb portion 402 and a lower limb portion 403 hinged at a hinged fold line position 407. The upper end of the upper limb portion 402 is hinged to the top plate 1, and the lower end of the lower limb portion 403 is hinged to the bottom plate 2. Figure 6As shown, the six limbs 4 are numbered L1, L2, L3, L4, L5, and L6 in a counter-clockwise direction. Limbs L1 and L4, when unfolded, have identical shapes and are both parallelograms. The remaining limbs 4 have different shapes when unfolded, with parallel sides but not vertically parallel top and bottom. During operation, the airbag 3 inflates and deflates, driving the top plate 1 and bottom plate 2 to move closer or separate. Simultaneously, each limb 4 unfolds or folds in coordination. When the actuator is fully extended, it is in a bent state to meet practical needs. Furthermore, during the unfolding or folding of the limbs 4, the invention achieves the coupling of torsional, telescopic, and bending movements, enabling bending motion and thus achieving diverse motion capabilities, expanding application scenarios. Both the top plate 1 and bottom plate 2 are regular hexagons.

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

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

[0064] like Figure 4 As shown, in this embodiment, the airbag 3 is a Krislin origami structure, which is a well-known technology in the art. The airbag 3 has an air nozzle 302 in the middle for connecting to the air tube for inflation and deflation. The airbag 3 has airbag connecting plates 301 at both the upper and lower ends for connecting to the top plate 1 and the bottom plate 2 respectively.

[0065] like Figure 5 As shown, in this embodiment, the upper end of the upper limb portion 402 of the limb 4 is provided with an upper hinge groove 405, and the lower end of the lower limb portion 403 is provided with a lower hinge groove 406. Furthermore, 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 5 to pass through.

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

[0067] The key to this invention lies in the folding position of the driving limb 4 to ensure that the actuator meets relevant design parameters, including bending angle and height. To simplify the design, the thicknesses of the limb 4, top plate 1, and bottom plate 2 are ignored. The specific design method of this invention includes the following steps:

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

[0069] like Figure 7 As shown, when the 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 invention is fully unfolded, the angle between the planes containing the top plate 1 and the bottom plate 2 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 top plate 1 and bottom plate 2. i .

[0071] To ensure the actuator has a bending effect, the present invention designs limbs 4, numbered L1 and L4, to form a parallelogram structure when unfolded. The remaining limbs 4, when unfolded, are only parallel on both sides, not vertically parallel. This... Figure 9 As shown, when the present invention is fully folded, the corner P of the top plate 1 corresponding to the limb 4 of serial number L1... u,1 and the corresponding base plate corner P d,1 Overlapping, the top plate corner P corresponding to limb 4 of serial number L4. u,4 and the corresponding base plate corner P d,4 They overlap, but after the actuator is fully deployed, due to the bending angle ζ, Figure 9 The remaining corners of the top plate 1 and the bottom plate 2 do not coincide in the projected view shown, and there is an included angle χ. i (i = 2, 3, 5, 6). By establishing a unit-length vector coordinate system, assuming clockwise is positive and counterclockwise is negative, the distance from the top corner 1 to the center O can be calculated using the formula for the angle between the two vectors. u Connect line r2 to the corner of base plate 2 and then to center O. d The angle between the lines r1 is:

[0072]

[0073] The 60° angle in equation (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 circumference.

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

[0075] θ i =θ b +χ i (2).

[0076] Step 3: When the invention is in the fully unfolded 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 points Ai, Bi, Ci, Di, Ei, Di′, Ei′, Fi, Gi and Hi of limb 4.

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

[0079] like Figure 8 As shown, when the invention is fully deployed, the equation of the plane containing the top plate 1 is:

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

[0081] Equation (3) above is obtained by referring to relevant mathematical formulas and is a well-known technique in this field, where y and z are... Figure 7 The coordinates of the coordinate system shown.

[0082] like Figure 10 As shown, assume that the corners of the i-th limb 4 after it is fully extended are Ai, Bi, Di′ and Ei′ respectively, and the corners of the i-th limb 4 after it is fully folded are Ai, Bi, Di and Ei respectively. At the same time, the two ends of the hinge line position 407 are Ci and Fi. A parallel line is drawn with Fi as the starting point, parallel to AiBi and intersecting BiCi at Hi. At the same time, the extension of DiEi and the extension of BiAi intersect at the origin O, and the intersection of the extension of DiEi and AiFi is Gi. Figure 10 This explanation will be based on the example of the i=2nd limb, 4.

[0083] Step 3.2: In the coordinate system with the center Od of 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 equations (4) and (5) above, Rot represents a rotational homogeneous coordinate transformation, Trans represents a translational homogeneous coordinate transformation, and l d，i like Figure 10 The figure shows the length of OAi, a. i like Figure 10 The figure shows the length of AiBi.

[0087] It can be represented as:

[0088]

[0089] In equation (6) above, αi is as follows Figure 10 The figure shows the angle between a perpendicular line drawn through Ai and AiFi.

[0090] Other vectors / / vector Di′ and Ei′ are respectively and The intersection point with the plane containing the top plate 1, therefore, by simultaneously solving equations (3) and (6), we can obtain the solution. and

[0091] but:

[0092]

[0093] In equations (7) and (8) above, as Figure 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 in this invention are completely identical, while the remaining limbs 4 are all different, and the definitions of each point are the same, please refer to [reference needed]. Figure 10 The limb 4 of L2 is shown. (As shown in the image) Figure 10 As shown, since A2F2 / / B2C2 (for the i-th limb, i.e., AiFi / / BiCi) and F2H2 / / A2B2 (FiHi / / AiBi), we have |F2H2|=a (i.e., |FiHi|=ai) and |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 triangle FiHiCi, by the Law of Sines:

[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 triangles OAiGi and FiEiGi:

[0099]

[0100] In equations (9) and (10) above, as Figure 10 As shown, λ i For ∠DiEiFi, ∠EiFiGi.

[0101] In triangle OGiAi, by the trigonometric sum of interior angles theorem, we have:

[0102]

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

[0104] Substituting equation (11) into equation (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: Solve equations (2), (7), (8), (9), and (12) simultaneously to form a system of constraint equations. Given the ai and α of the i-th limb... i θ i When parameters such as ζ are obtained, the size of the limb 4 can be determined by solving the problem using computer software. 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: It includes a top plate (1), a bottom plate (2), an airbag (3), and limbs (4). 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 around the outside of the airbag (3) and located between the top plate (1) and the bottom plate (2). Each limb (4) includes an upper limb (402) and a lower limb (403) hinged at the hinge fold line position (407). The upper limb (402) has an upper limb part (402) and a lower limb part (403) hinged at the hinge fold line position (407). The lower end of the lower limb (403) is hinged to the top plate (1), and the lower end of the lower limb (403) is hinged to the bottom plate (2). The six limbs (4) are numbered L1, L2, L3, L4, L5, and L6 in a counterclockwise direction. The limbs (4) numbered L1 and L4 have the same shape after unfolding and are all parallelograms. The other limbs (4) have different shapes after unfolding and are all parallel on both sides but not parallel up and down. 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 top plate (1) is provided with top plate connecting holes (101) and top plate hinge seats (102) interlaced along the circumferential direction on the lower side, wherein the upper end of the upper body part (402) is hinged to the corresponding top plate hinge seat (102) by a pin (5).

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

4. The pneumatic bistable coupled motion actuator according to claim 1, characterized in that: The airbag (3) has an air nozzle (302) in the middle, and the airbag (3) has an airbag connecting plate (301) at both the upper and lower ends.

5. The pneumatic bistable coupled motion actuator according to claim 1, characterized in that: The upper limb (402) of the limb (4) is provided with an upper hinge groove (405) at the upper end and the lower limb (403) is provided with a lower hinge groove (406) at the lower end. 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) through which the pin (5) passes.

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

7. A design method for a pneumatic bistable coupled motion actuator according to claim 1, characterized in that: Includes the following steps: Step 1: Define the torsion angle θ of the top plate (1) and the bottom plate (2). b The bending angle ζ and the height H, where 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 plane containing the top plate (1) and the bottom plate (2) is the bending angle ζ, and at the same time, the center O of the top plate (1) u and the center O of the base 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 P of the top plate (1) corresponding to the limb (4) of serial number L1. u，1 The corner end Pd,1 of the corresponding base plate (2) coincides with the corner end Pu,4 of the corresponding top plate (1) of the limb (4) of serial number L4, and the corner end Pd,4 of the corresponding base plate (2) coincides with the corner end Pd,4. The remaining corner ends of the top plate (1) and the remaining corner ends of the base plate (2) do not coincide, and there is an included angle χ. i Let i = 1, 2, ..., 6. Assuming clockwise is positive and counterclockwise is negative, according to the formula for the angle between vectors: Actual torsion angle θ i for: i i =θ b +x i (2); Step 3: Determine the parameter relationship between the bending angle ζ and the limb (4) when fully extended, specifically: Step 3.1: Establish a coordinate system and determine the positions of each point A of the limbs (4). i B i C i D i E i D i E′ i 、F i G i and H i Among them, the center O of the base plate (2) d Establish a coordinate system with the origin O and coordinates x, y, z. The equation of the plane containing the top plate (1) when fully unfolded is: -sin(ξ)y+cos(ξ)z-cos(ξ)H=0 (3); Let A be the corners of the i-th limb (4) after it is fully extended. i B i 、D′ i and E′ i After the i-th limb (4) is fully folded, each corner end is A i B i 、D i and E i Meanwhile, the two ends of the hinged broken line position (407) are C i 、F i , with F i Draw a line parallel to A as the starting point. i B i Parallel and with B i C i The intersection point is H i Meanwhile, D i E i Extension line and B i A i The extensions of the lines intersect at the origin O, and D i E i Extension line and A i F i The intersection point is G i ; Step 3.2: At the center O of the base plate (2) d In a coordinate system with origin O, using homogeneous coordinate transformation, A i B i 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°+x i )Trans(x,l d，i +a i )[0 0 0 1] T (5); In equations (4) and (5) above, Rot represents a rotational homogeneous coordinate transformation, Trans represents a translational homogeneous coordinate transformation, and l d，i For OA i The length of A, ai is A i B i length; Expressed as: In equation (6) above, α i For A i Draw a perpendicular line and make the perpendicular line intersect A. i F i The angle between them; Other vectors / / vector D′ i E′ i At the same time respectively and The intersection point with the plane containing the top plate (1), therefore, by simultaneously solving equations (3) and (6), we can obtain the solution. and but: In equations (7) and (8) above, b i For B i C i Length, ci is C i D′ i Length, ei is E′ i F i Length, f i For F i A i length; Step 4: Determine the position of the hinge fold line (407) and the torsion angle θ when fully folded. i The relationship is as follows: for the i-th limb (4), in triangle F i H i C i In the middle, by the Law of Sines, we get: (b) i -f i )cos(α i -b i )=a i sin(β i ) (9); Similarly, in triangle OA i G i Neutral triangle F i E i G i The following relationship is obtained: In equations (9) and (10) above, λ i For ∠D i E i F i , For ∠E i F i G i ; In triangle OG i A i In the triangle and interior angle sum theorem, we have: In equation (11) above, θ i For ∠A i OG i ; Substituting equation (11) into equation (10), we get: f i cos(a i -θ i )-l d，i sin(θ i )=e i sin(λ i ) (12); Step 5: Solve the equations (2), (7), (8), (9) and (12) to form 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 computer software based on the above set of constraint equations.

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