Method and system for calculating degrees of freedom of multi-loop coupling mechanisms based on geometric algebra

Abstract

The present invention provides a method and system for calculating the degrees of freedom of a multi-loop coupling mechanism based on geometric algebra, including: determining the basic branch of a Twin-Bennett mechanism to obtain the initial motion space of the basic branch; finding all coupled branches that form a closed loop with the basic branch and calculating the branch motion space of the coupled branch; judging whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop among all basic branches containing the closed loop; if so, equating the motion space of the basic branch in the closed loop to the motion space of the closed loop; if not, calculating the intersection of the motion spaces of the basic branch and the coupled branch in the closed loop to be equivalent to the motion space of the new basic branch; finding the intersection of the motion spaces of all branches of the equivalent mechanism to obtain the motion space of the output platform, thereby deriving the degrees of freedom of the Twin-Bennett mechanism. The present invention has a simple analysis and calculation process and clear physical meaning.

Description

Technical Field

[0001] The present invention relates to the technical field of degree of freedom calculation of a multi-ring coupling mechanism, and specifically, to a method and system for calculating the degree of freedom of a multi-ring coupling mechanism based on geometric algebra, and more specifically, to a method and system for analyzing the degree of freedom of a multi-ring coupling mechanism based on equivalent branches of geometric algebra. Background Art

[0002] Calculating the degrees of freedom of a mechanism is fundamental to the analysis and design of new machines. Degrees of freedom are one of the most important properties of a mechanism, and their calculation or analysis is a fundamental issue in the study of mechanisms. For a long time, the GK formula has been used to calculate degrees of freedom. However, with the development of mechanisms, many counterexamples have emerged, such as in some classical mechanisms like Bennett and Bricard, where the classic GK formula cannot accurately produce accurate results. For over 150 years, numerous scholars have conducted research on calculating the degrees of freedom of mechanisms. Huang Zhen proposed a modified GK formula for calculating degrees of freedom. This method, based on screw theory, establishes the kinematic screw of the mechanism's kinematic pairs. The reciprocal product of the screw is calculated to obtain the anti-screw. Based on the anti-screw, the overconstraints and common constraints are analyzed, and finally the modified GK formula is used for calculation. This method has the advantages of a simple analysis process, clear physical meaning, and the ability to obtain the specific kinematic mode of the mechanism, leading to its widespread application. However, its disadvantage is that the calculation of the anti-screw twice is required to determine the constraint screw, making the calculation process cumbersome for complex multi-loop coupled mechanisms, such as the Hoberman switch-pitch mechanism and the deformable Rubik's cube mechanism. Gogu initially proposed a method for calculating degrees of freedom by solving the rank of the kinematic chain constraint equations. This method is effective for most parallel mechanisms, but is more difficult for complex ones. Later, he proposed incorporating the number of rods and kinematic pairs into the formula for calculating degrees of freedom. This method does not require the establishment of closed-loop equations for the kinematic chain, and the process is simple, but it is not applicable to all mechanisms. This includes classical mechanisms such as Bennett and Goldberg, as well as modern parallel mechanisms such as Delta and H4.

[0003] In recent years, the structure of the mechanism has begun to develop towards a more complex spatial structure, and the parallel mechanism has gradually developed from a simple open-loop branch chain to a multi-loop coupling mechanism, such as some folding and unfolding mechanisms and magic cube mechanisms. The traditional degree of freedom calculation method has the disadvantages of complex analysis process and cumbersome calculation process when dealing with these mechanisms. Therefore, it is very necessary to propose an efficient and concise method for calculating the degree of freedom of multi-loop coupling mechanisms.

[0004] Patent document CN102393876A (application number: CN201110195285.5) discloses a method for calculating the degrees of freedom of a parallel mechanism using the degrees of freedom of a rod group. Based on the principle of minimizing the base point motion parameters in the parallel mechanism, a fixed coordinate system is established on the frame. The number of degrees of freedom of the rod group kinematic pairs, as well as the rod group motion parameters Gk and dimension are calculated. The difference between the two is used to calculate the rod group's degrees of freedom Fk. The sum of the dimensions of the output component base point motion parameters is then used to calculate the parallel mechanism's degrees of freedom F. However, this invention does not address the geometric algebraic method for calculating the degrees of freedom of multi-loop coupling mechanisms. Summary of the Invention

[0005] In view of the defects in the prior art, the purpose of the present invention is to provide a method and system for calculating the degrees of freedom of a multi-loop coupling mechanism based on geometric algebra.

[0006] According to the present invention, a method for calculating the degrees of freedom of a multi-ring coupling mechanism based on geometric algebra is provided, comprising:

[0007] Step S1: Determine the basic branch chain of the Twin-Bennett mechanism and obtain the initial motion space of the basic branch chain;

[0008] Step S2: Find all coupled branches that form a closed loop with the base branch and calculate the branch motion space of the coupled branches;

[0009] Step S3: determining whether the motion space of the coupled branch in the closed loop among all the basic branches containing the closed loop belongs to the motion space of the basic branch in the closed loop;

[0010] If yes, the motion space of the basic branch in the closed loop is equivalent to the motion space of the closed loop;

[0011] If it does not belong to the category, the intersection of the motion space of the basic branch and the coupled branch in the closed loop is calculated to be equivalent to the motion space of the new basic branch;

[0012] Step S4: The motion spaces of all branches of the equivalent mechanism are intersected to obtain the motion space of the output platform, thereby obtaining the degrees of freedom of the Twin-Bennett mechanism.

[0013] Preferably, in step S1:

[0014] Select the fixed platform and the moving platform of the mechanism, divide the basic branches, write the motion spiral of the kinematic pair on each basic branch, calculate the branch motion space of each basic branch, and obtain the initial motion space of the basic branch chain;

[0015] The spiral is represented in six-dimensional geometric algebra G6 as a one-dimensional piece product:

[0016] S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6

[0017] where e1, e2, …, e6 represent the basis in G6, and the scalar coefficient v i and b i , i = 1, 2, 3, represents the Plücker coordinates of the secondary axis of motion;

[0018] The slice product S formed by the outer product operation of all n motion spirals on the i-th branch mi Represents the motion space of this branch:

[0019] S mi =S i1 ∧S i2 ∧…∧S in

[0020] Among them S ij (j=1,…,n) represents the motion spiral of the j-th motion pair on the i-th branch, and ∧ represents the outer product operation symbol.

[0021] Preferably, in step S2:

[0022] Find all coupled branches that form a closed loop with the base branch, write the kinematic spiral of the kinematic pair on each coupled branch, and calculate the branch motion space of the coupled branch;

[0023] The branch motion space S' of the coupled branch chain coupled with the i-th basic branch in the closed loop mij Expressed as:

[0024] S' mij =S' ij1 ∧S' ij2 ∧…∧S' ijn

[0025] where S' mij (j=1,…,n) represents the motion space of the jth coupled branch coupled with the i-th basic branch, S' ijk (k=1,…,n) represents the motion spiral of the kth motion pair on the jth closed-loop branch coupled with the i-th basic branch.

[0026] Preferably, in step S3:

[0027] Determine whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop. The determination method is: perform an outer product operation on the motion space of each branch in the closed loop, and determine whether the motion space of one branch belongs to the motion space of another branch based on the operation result;

[0028] If the result of the outer product operation is 0, it means that the motion space of the coupled branch chain coupled with the basic branch in the closed loop belongs to the motion space of the basic branch. The coupled branch chain in the closed loop has no effect on the motion of the basic branch chain. Then the motion space S of the basic branch chain in the closed loop is converted to mi The motion space is equivalent to a closed loop;

[0029] If the result of the outer product operation is not 0, the motion space S of the basic branch mi and the motion space S' of the coupled branches coupled to the base branch mi1 , S' mi2 ,…,S' mij Find the intersection and use this intersection to replace the motion space of the closed-loop output.

[0030] Preferably, in step S4:

[0031] The method for finding the intersection is: when the union of the allowable subspaces at the ends of the two branch kinematic chains is I6, the intersection S mM for:

[0032]

[0033] Where I6 represents the maximum order piece product in 6-dimensional geometric algebra, which is composed of the outer product of 6 unit vectors, I6=e1∧e2∧…∧e6, Yes, I6 inverse,

[0034] When the union of the permissible subspaces at the ends of two branch kinematic chains is not equal to I6, the union of the two, I6, is used. u Instead of I6, the intersection S mM for:

[0035]

[0036] According to the present invention, a multi-loop coupling mechanism degree of freedom calculation system based on geometric algebra is provided, comprising:

[0037] Module M1: Determine the basic branch of the Twin-Bennett mechanism and obtain the initial motion space of the basic branch;

[0038] Module M2: Find all coupled branches that form a closed loop with the base branch and calculate the branch motion space of the coupled branches;

[0039] Module M3: Determine whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop among all the basic branches containing the closed loop;

[0040] If yes, the motion space of the basic branch in the closed loop is equivalent to the motion space of the closed loop;

[0041] If it does not belong to the category, the intersection of the motion space of the basic branch and the coupled branch in the closed loop is calculated to be equivalent to the motion space of the new basic branch;

[0042] Module M4: Intersect the motion spaces of all branches of the equivalent mechanism to obtain the motion space of the output platform, thereby obtaining the degrees of freedom of the Twin-Bennett mechanism.

[0043] Preferably, in the module M1:

[0044] Select the fixed platform and the moving platform of the mechanism, divide the basic branches, write the motion spiral of the kinematic pair on each basic branch, calculate the branch motion space of each basic branch, and obtain the initial motion space of the basic branch chain;

[0045] The spiral is represented in six-dimensional geometric algebra G6 as a one-dimensional piece product:

[0046] S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6

[0047] where e1, e2, …, e6 represent the basis in G6, and the scalar coefficient v i and b i , i = 1, 2, 3, represents the Plücker coordinates of the secondary axis of motion;

[0048] The slice product S formed by the outer product operation of all n motion spirals on the i-th branch mi Represents the motion space of this branch:

[0049] S mi =S i1 ∧S i2 ∧…∧S in

[0050] Among them S ij (j=1,…,n) represents the motion spiral of the j-th motion pair on the i-th branch, and ∧ represents the outer product operation symbol.

[0051] Preferably, in the module M2:

[0052] Find all coupled branches that form a closed loop with the base branch, write the kinematic spiral of the kinematic pair on each coupled branch, and calculate the branch motion space of the coupled branch;

[0053] The branch motion space S' of the coupled branch chain coupled with the i-th basic branch in the closed loop mij Expressed as:

[0054] S' mij =S' ij1 ∧S' ij2 ∧…∧S' ijn

[0055] where S' mij (j=1,…,n) represents the motion space of the jth coupled branch coupled with the i-th basic branch, S' ijk (k=1,…,n) represents the motion spiral of the kth motion pair on the jth closed-loop branch coupled with the i-th basic branch.

[0056] Preferably, in the module M3:

[0057] Determine whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop. The determination method is: perform an outer product operation on the motion space of each branch in the closed loop, and determine whether the motion space of one branch belongs to the motion space of another branch based on the operation result;

[0058] If the result of the outer product operation is 0, it means that the motion space of the coupled branch chain coupled with the basic branch in the closed loop belongs to the motion space of the basic branch. The coupled branch chain in the closed loop has no effect on the motion of the basic branch chain. Then the motion space S of the basic branch chain in the closed loop is converted to mi The motion space is equivalent to a closed loop;

[0059] If the result of the outer product operation is not 0, the motion space S of the basic branch mi and the motion space S' of the coupled branches coupled to the base branch mi1 , S' mi2 ,…,S' mij Find the intersection and use this intersection to replace the motion space of the closed-loop output.

[0060] Preferably, in the module M4:

[0061] The method for finding the intersection is: when the union of the allowable subspaces at the ends of the two branch kinematic chains is I6, the intersection S mM for:

[0062]

[0063] Where I6 represents the maximum order piece product in 6-dimensional geometric algebra, which is composed of the outer product of 6 unit vectors, I6=e1∧e2∧…∧e6, Yes, I6 inverse,

[0064] When the union of the permissible subspaces at the ends of two branch kinematic chains is not equal to I6, the union of the two, I6, is used. u Instead of I6, the intersection S mM for:

[0065]

[0066] Compared with the prior art, the present invention has the following beneficial effects:

[0067] 1. The analysis and calculation process of the present invention is simple. When dealing with a multi-ring coupling mechanism, the degree of freedom calculation method of the present invention only needs to determine the motion space of the branches in the closed loop and use the equivalent branch method to analyze and process, which is concise and easy to understand. Compared with the use of spiral theory to solve the anti-spiral system, the present invention uses the mathematical tool of geometric algebra, which only involves addition and multiplication in the calculation. It can not only perform numerical solutions, but also simplifies the solution of symbolic expressions of the motion space.

[0068] 2. The physical meaning of the present invention is clear. It is calculated under the framework of geometric algebra, represents the motion of the branch chain in , and represents the motion space through intersection calculation. The geometric meaning of the multi-order piece product is clear. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Other features, objects and advantages of the present invention will become more apparent upon reading the detailed description of non-limiting embodiments with reference to the following drawings:

[0070] Figure 1 Flowchart for calculating the degree of freedom of the present invention

[0071] Figure 2 Schematic diagram of the Twin-Bennett mechanism according to an embodiment of the present invention.

[0072] Figure 3 2 is a Twin-Bennett mechanism diagram of an embodiment of the present invention.

[0073] Figure 4 This is a joint coordinate system position diagram of an embodiment of the present invention.

[0074] Figure 5 2 is a motion spiral diagram of the Twin-Bennett mechanism according to an embodiment of the present invention. DETAILED DESCRIPTION

[0075] The present invention will be described in detail below with reference to specific embodiments. The following examples will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, for those skilled in the art, several changes and improvements can be made without departing from the scope of the present invention. These all fall within the scope of protection of the present invention.

[0076] Example 1:

[0077] The present invention provides a method for calculating the degree of freedom of a multi-ring coupling mechanism based on geometric algebra, comprising: selecting a fixed platform and a movable platform of the mechanism, dividing the basic branch and the coupled branch coupled thereto in a closed loop, writing the motion spiral of the kinematic pair on each branch, and calculating the motion space of each branch; judging the relationship between the motion space of the coupled branch and the basic branch, and equivalently converting it into a new motion space; performing the above steps on all basic branches containing closed loops; calculating the intersection of the motion spaces of all equivalent branches, and obtaining the motion space of the output platform, thereby obtaining the degree of freedom of the mechanism. The method for calculating the degree of freedom of the present invention only needs to determine the motion spiral of each branch of the multi-ring coupling mechanism to calculate the branch motion space, and then obtain the degree of freedom of the mechanism's movable platform, which can conveniently and concisely calculate the degree of freedom of the multi-ring coupling mechanism.

[0078] According to the present invention, a method for calculating the degree of freedom of a multi-ring coupling mechanism based on geometric algebra is provided. Figure 1-Figure 5 Shown include:

[0079] Step S1: Determine the basic branch chain of the Twin-Bennett mechanism and obtain the initial motion space of the basic branch chain;

[0080] Specifically, in step S1:

[0081] Select the fixed platform and the moving platform of the mechanism, divide the basic branches, write the motion spiral of the kinematic pair on each basic branch, calculate the branch motion space of each basic branch, and obtain the initial motion space of the basic branch chain;

[0082] The spiral is represented in six-dimensional geometric algebra G6 as a one-dimensional piece product:

[0083] S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6

[0084] where e1, e2, …, e6 represent the basis in G6, and the scalar coefficient v i and b i , i = 1, 2, 3, represents the Plücker coordinates of the secondary axis of motion;

[0085] The slice product S formed by the outer product operation of all n motion spirals on the i-th branch mi Represents the motion space of this branch:

[0086] S mi =S i1 ∧S i2 ∧…∧S in

[0087] Among them S ij (j=1,…,n) represents the motion spiral of the j-th motion pair on the i-th branch, and ∧ represents the outer product operation symbol.

[0088] Step S2: Find all coupled branches that form a closed loop with the base branch and calculate the branch motion space of the coupled branches;

[0089] Specifically, in step S2:

[0090] Find all coupled branches that form a closed loop with the base branch, write the kinematic spiral of the kinematic pair on each coupled branch, and calculate the branch motion space of the coupled branch;

[0091] The branch motion space S' of the coupled branch chain coupled with the i-th basic branch in the closed loop mij Expressed as:

[0092] S' mij =S' ij1 ∧S' ij2 ∧…∧S' ijn

[0093] where S' mij (j=1,…,n) represents the motion space of the jth coupled branch coupled with the i-th basic branch, S' ijk (k=1,…,n) represents the motion spiral of the kth motion pair on the jth closed-loop branch coupled with the i-th basic branch.

[0094] Step S3: determining whether the motion space of the coupled branch in the closed loop among all the basic branches containing the closed loop belongs to the motion space of the basic branch in the closed loop;

[0095] If yes, the motion space of the basic branch in the closed loop is equivalent to the motion space of the closed loop;

[0096] If it does not belong to the category, the intersection of the motion space of the basic branch and the coupled branch in the closed loop is calculated to be equivalent to the motion space of the new basic branch;

[0097] Specifically, in step S3:

[0098] Determine whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop. The determination method is: perform an outer product operation on the motion space of each branch in the closed loop, and determine whether the motion space of one branch belongs to the motion space of another branch based on the operation result;

[0099] If the result of the outer product operation is 0, it means that the motion space of the coupled branch chain coupled with the basic branch in the closed loop belongs to the motion space of the basic branch. The coupled branch chain in the closed loop has no effect on the motion of the basic branch chain. Then the motion space S of the basic branch chain in the closed loop is converted to mi The motion space is equivalent to a closed loop;

[0100] If the result of the outer product operation is not 0, the motion space S of the basic branch mi and the motion space S' of the coupled branches coupled to the base branch mi1 , S' mi2 ,…,S' mij Find the intersection and use this intersection to replace the motion space of the closed-loop output.

[0101] Step S4: The motion spaces of all branches of the equivalent mechanism are intersected to obtain the motion space of the output platform, thereby obtaining the degrees of freedom of the Twin-Bennett mechanism.

[0102] Specifically, in step S4:

[0103] The method for finding the intersection is: when the union of the allowable subspaces at the ends of the two branch kinematic chains is I6, the intersection S mM for:

[0104]

[0105] Where I6 represents the maximum order piece product in 6-dimensional geometric algebra, which is composed of the outer product of 6 unit vectors, I6=e1∧e2∧…∧e6, Yes, I6 inverse,

[0106] When the union of the permissible subspaces at the ends of two branch kinematic chains is not equal to I6, the union of the two, I6, is used. u Instead of I6, the intersection S mM for:

[0107]

[0108] According to the present invention, a multi-loop coupling mechanism degree of freedom calculation system based on geometric algebra is provided, comprising:

[0109] Module M1: Determine the basic branch of the Twin-Bennett mechanism and obtain the initial motion space of the basic branch;

[0110] Specifically, in the module M1:

[0111] Select the fixed platform and the moving platform of the mechanism, divide the basic branches, write the motion spiral of the kinematic pair on each basic branch, calculate the branch motion space of each basic branch, and obtain the initial motion space of the basic branch chain;

[0112] The spiral is represented in six-dimensional geometric algebra G6 as a one-dimensional piece product:

[0113] S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6

[0114] where e1, e2, …, e6 represent the basis in G6, and the scalar coefficient v i and b i , i = 1, 2, 3, represents the Plücker coordinates of the secondary axis of motion;

[0115] The slice product S formed by the outer product operation of all n motion spirals on the i-th branch mi Represents the motion space of this branch:

[0116] S mi =S i1 ∧S i2 ∧…∧S in

[0117] Among them S ij (j=1,…,n) represents the motion spiral of the j-th motion pair on the i-th branch, and ∧ represents the outer product operation symbol.

[0118] Module M2: Find all coupled branches that form a closed loop with the base branch and calculate the branch motion space of the coupled branches;

[0119] Specifically, in the module M2:

[0120] Find all coupled branches that form a closed loop with the base branch, write the kinematic spiral of the kinematic pair on each coupled branch, and calculate the branch motion space of the coupled branch;

[0121] The branch motion space S' of the coupled branch chain coupled with the i-th basic branch in the closed loop mij Expressed as:

[0122] S' mij =S' ij1 ∧S' ij2 ∧…∧S' ijn

[0123] where S' mij (j=1,…,n) represents the motion space of the jth coupled branch coupled with the i-th basic branch, S' ijk (k=1,…,n) represents the motion spiral of the kth motion pair on the jth closed-loop branch coupled with the i-th basic branch.

[0124] Module M3: Determine whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop among all the basic branches containing the closed loop;

[0125] If yes, the motion space of the basic branch in the closed loop is equivalent to the motion space of the closed loop;

[0126] If it does not belong to the category, the intersection of the motion space of the basic branch and the coupled branch in the closed loop is calculated to be equivalent to the motion space of the new basic branch;

[0127] Specifically, in the module M3:

[0128] Determine whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop. The determination method is: perform an outer product operation on the motion space of each branch in the closed loop, and determine whether the motion space of one branch belongs to the motion space of another branch based on the operation result;

[0129] If the result of the outer product operation is 0, it means that the motion space of the coupled branch chain coupled with the basic branch in the closed loop belongs to the motion space of the basic branch. The coupled branch chain in the closed loop has no effect on the motion of the basic branch chain. Then the motion space S of the basic branch chain in the closed loop is converted to mi The motion space is equivalent to a closed loop;

[0130] If the result of the outer product operation is not 0, the motion space S of the basic branch mi and the motion space S' of the coupled branches coupled to the base branch mi1 , S' mi2 ,...,S' mij Find the intersection and use this intersection to replace the motion space of the closed-loop output.

[0131] Module M4: Intersect the motion spaces of all branches of the equivalent mechanism to obtain the motion space of the output platform, thereby obtaining the degrees of freedom of the Twin-Bennett mechanism.

[0132] Specifically, in the module M4:

[0133] The method for finding the intersection is: when the union of the allowable subspaces at the ends of the two branch kinematic chains is I6, the intersection S mM for:

[0134]

[0135] Where I6 represents the maximum order piece product in 6-dimensional geometric algebra, which is composed of the outer product of 6 unit vectors, I6=e1∧e2∧...∧e6, Yes, I6 inverse,

[0136] When the union of the permissible subspaces at the ends of two branch kinematic chains is not equal to I6, the union of the two, I6, is used. u Instead of I6, the intersection S mM for:

[0137]

[0138] Example 2:

[0139] Example 2 is a preferred example of Example 1 and is used to illustrate the present invention in more detail.

[0140] This paper proposes a method for calculating degrees of freedom using equivalent branches. This method uses geometric algebra to calculate the motion spaces of the closed-loop branches. The outer product of the motion spaces of each branch in the closed loop is then calculated to determine the linear correlation between the two branches. If the outer product is zero, indicating that the motion spaces of the two branches are linearly correlated, one of the branches can be used to equivalently replace the closed-loop structure for degree of freedom calculation. If the outer product is not zero, the intersection of the motion spaces of the two branches in the closed loop is calculated and used to replace the motion space output by the closed loop. Finally, the intersection of the motion spaces of all equivalent branches is calculated to determine the degrees of freedom of the mechanism's dynamic platform.

[0141] In recent years, space technology has developed rapidly. For example, large-scale space solar power stations, the International Space Station, and space planes, etc., the solar panels, antennas and other solid-surface deployment mechanisms used on them belong to a type of deployable mechanism, which is a typical multi-ring coupling mechanism.

[0142] The Twin-Bennett mechanism is a dual-loop coupling mechanism ( Figure 2 (Figure 1 is a simplified diagram of the mechanism) and can serve as the basic building block of a deployable mechanism. Foldable mechanisms are typically assembled from multiple repetitive units, most of which are over-constrained mechanisms. The Twin-Bennett mechanism consists of two identical Bennett mechanisms. As a typical over-constrained mechanism, the Bennett mechanism can provide high strength without causing significant strain. For example, a foldable solar concentrator with the Twin-Bennett mechanism as its basic unit can reduce the volume of the concentrator in its non-operating state through the foldable nature of the Twin-Bennett mechanism. Therefore, it can be applied to the effective reuse of solar energy in space, and has very important practical significance for tasks such as spacecraft propulsion and attitude adjustment.

[0143] Take two identical Bennett mechanisms, which are arranged symmetrically and connected by a common link and kinematic pair to form an over-constrained 6R mechanism, forming a Twin-Bennett mechanism ( Figure 2 (Figure 2 is a simplified diagram of the mechanism.) This mechanism contains six revolute pairs and seven connecting rods. Due to the properties of the Bennett link, the opposing connecting rods are identical and have the same torsion angle. The Twin-Bennett mechanism satisfies the following geometric relationships:

[0144] OA=CB=CD=a

[0145] OC=AB=AD=b

[0146] αOA =α CB =α1=-α,α CD =α2=α

[0147] α OC =α AB =α AD =β

[0148]

[0149]

[0150]

[0151] Take the center point of the revolving joint 6 as the origin to establish Figure 5 The fixed coordinate system is shown in the figure, where the z-axis is along the axis rod direction of the revolute pair 6, the angle between the d-rod and the x-axis is θ, α is the torsion angle of the axis of the revolute pairs 3 and 4 about the z-axis, and β is the torsion angle between the revolute pairs 1 and 6. represents the angle between rod c and rod f, represents the angle between rod c and rod b, and m and l represent the corresponding rod lengths.

[0152] The dynamic platform and the fixed platform are determined as follows Figure 3 As shown, Figure 4 As shown, the two basic branches are:

[0153]

[0154] The motion space of the two basic branches can be written as:

[0155] S m1 =S 11 ∪S 12

[0156] S m2 =S 21 ∪S 22

[0157] The coupling branch CDA is coupled with the basic branch COA to form a closed loop, such as Figure 4 As shown in the figure, the motion spirals of the four revolute pairs on the closed loop are expressed in the geometric algebra framework as follows:

[0158] S l11 =e3

[0159] S l21 =-sαe2+cαe3-acαe5-asαe6

[0160]

[0161] Sl23 =-sθsβe1+cθsβe2+cβe3+bsθcβe4-bcθcβe5+bsβe6

[0162] The motion space of the two branches on the closed loop is:

[0163] S l1 =S l11 =e3

[0164] S l2 =S l21 ∧S l22 ∧S l23

[0165] =sαsθsβe 123 -acαsθsβe 135 -asαsθsβe 1,3,6 -sαsθbcβe 234 +(acαcθsβ+bsαcθcβ)e 235 +(asαcθsβ-sαsβb)e 236 -abcαsθcβe 345 -absαsθcβe 346 +(absαcθcβ+abcαsβ)e 356

[0166] Two branches S l1 , S l2 The outer product operation results in:

[0167] S l1 ∧S l2 =0

[0168] The outer product result is 0, indicating that S l1 , S l2 is linearly related, S l1 ∩S l2 =S l1 , which has no effect on the motion space of the coupled basic branch chain, and the motion space of the closed loop can be equivalent to S1.

[0169] The motion spirals of the revolute pairs on the two basic branches are:

[0170] S 11 =e3

[0171] S 12 =-sθsβe1+cθsβe2+cβe3+bsθcβe4-bcθcβe5+bsβe6

[0172] S 21 =sαe2+cαe3-acαe5-asαe6

[0173]

[0174] The motion space of the two basic branches is:

[0175] S m1 = S 11 ∧ S 12 = sθsβe 13 -cθsβe 23 +bsθcβe 34 -bcθcβe 35 +bsβe 36

[0176]

[0177] The intersection of the motion spaces of the two branches is:

[0178]

[0179] J = S m1 ∪ S m2

[0180] = S 11 ∧ S 12 ∧ S 22

[0181] = sαsθsβe 123 +acαsθsβe 135 +asαsθsβe 136 -sαsθbcβe 234 +(bsαcθcβ - acαcθcβ)e 235 -(asαcθsβ + sαsβb)e 236 +abcαsθcβe 345 +absαsθcβe 346 -ab(sαcθcβ + cαsβ)e 356

[0182] Where C is:

[0183] c1 = (a 2 b 2 +1)c 2 βc 2 θc 2 α - 2a 2 b 2 cθcαsαcβ + 2abc 2 θcαsαsβ + a 2 b 2 c 2 βc 2α-2abcθc 2 βc 2 α-a 2 b 2 c 2 α-a 2 b 2 c 2 β+2abcθc 2 α+2abcθc 2 β-c 2 θc 2 α-c 2 θc 2 β-c 2 βc 2 α+b 2 c 2 α+a 2 c 2 β-2abcθ+c 2 θ+c 2 α+c 2 β-a 2 -b 2 -1

[0184] The result of the motion space of the moving platform is a 1-order piece product, which means that the moving platform of the mechanism has 1 degree of freedom. The angle θ is the only variable. The motion spiral of the moving platform under different configurations is as follows: Figure 5 As shown in the figure, line ① represents the direction of the permissible spiral on the moving platform under different configurations, line ② represents the trace of a point on the permissible spiral, and the moving platform is in full-circle motion.

[0185] Those skilled in the art will appreciate that, in addition to implementing the system, device, and various modules provided by the present invention in purely computer-readable program code, it is entirely possible to implement the same program in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, embedded microcontrollers, and the like by logically programming the method steps. Therefore, the system, device, and various modules provided by the present invention can be considered a hardware component, and the modules included therein for implementing various programs can also be considered structures within the hardware component; the modules for implementing various functions can also be considered both software programs for implementing the method and structures within the hardware component.

[0186] The above describes specific embodiments of the present invention. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art may make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. The embodiments of this application and the features in the embodiments may be combined with each other in any manner unless there is a conflict.

Claims

1. A method for calculating the degrees of freedom of a multi-loop coupling mechanism based on geometric algebra, characterized in that: include: Step S1: Determine the basic branch chain of the Twin-Bennett mechanism and obtain the initial motion space of the basic branch chain; Step S2: Find all coupled branches that form a closed loop with the base branch and calculate the branch motion space of the coupled branches; Step S3: determining whether the motion space of the coupled branch in the closed loop among all the basic branches containing the closed loop belongs to the motion space of the basic branch in the closed loop; If yes, the motion space of the basic branch in the closed loop is equivalent to the motion space of the closed loop; If it does not belong to the category, the intersection of the motion space of the basic branch and the coupled branch in the closed loop is calculated to be equivalent to the motion space of the new basic branch; Step S4: finding the intersection of the motion spaces of all branches of the equivalent mechanism to obtain the motion space of the output platform, thereby obtaining the degrees of freedom of the Twin-Bennett mechanism; In step S3: Determine whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop. The determination method is: perform an outer product operation on the motion space of each branch in the closed loop, and determine whether the motion space of one branch belongs to the motion space of another branch based on the operation result; If the result of the outer product operation is 0, it means that the motion space of the coupled branch chain coupled with the basic branch in the closed loop belongs to the motion space of the basic branch. The coupled branch chain in the closed loop has no effect on the motion of the basic branch chain. Then the motion space S of the basic branch chain in the closed loop is converted to mi The motion space is equivalent to a closed loop; If the result of the outer product operation is not 0, the motion space S of the basic branch mi and the motion space S′ of the coupled branches coupled to the base branch mi1 , S′ mi2 , L, S′ mij Find the intersection and use this intersection to replace the motion space of the closed-loop output; In step S4: The method for finding the intersection is: when the union of the allowable subspaces at the ends of the two branch kinematic chains is I6, the intersection S mM for: Where I6 represents the maximum order piece product in 6-dimensional geometric algebra, which is composed of the outer product of 6 unit vectors, I6=e1∧e2∧L∧e6, Yes, I6 inverse, When the union of the permissible subspaces at the ends of two branch kinematic chains is not equal to I6, the union of the two, I6, is used. u Instead of I6, the intersection S mM for:

2. The method for calculating the degree of freedom of a multi-loop coupling mechanism based on geometric algebra according to claim 1, characterized in that: In step S1: Select the fixed platform and the moving platform of the mechanism, divide the basic branches, write the motion spiral of the kinematic pair on each basic branch, calculate the branch motion space of each basic branch, and obtain the initial motion space of the basic branch chain; The spiral is represented in six-dimensional geometric algebra G6 as a one-dimensional piece product: S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6 where e1, e2, L, e6 represent the basis in G6, and the scalar coefficient v i and b i , i = 1, 2, 3, represents the Plücker coordinates of the secondary axis of motion; The slice product S formed by the outer product operation of all n motion spirals on the i-th branch mi Represents the motion space of this branch: S mi =S i1 ∧S i2 ∧L∧S in Among them S ij (j=1, L, n) represents the motion spiral of the j-th motion pair on the i-th branch, and ∧ represents the outer product operation symbol.

3. The method for calculating the degree of freedom of a multi-loop coupling mechanism based on geometric algebra according to claim 1, characterized in that: In step S2: Find all coupled branches that form a closed loop with the base branch, write the kinematic spiral of the kinematic pair on each coupled branch, and calculate the branch motion space of the coupled branch; The branch motion space S′ of the coupled branch chain coupled with the i-th basic branch in the closed loop mij Expressed as: S′ mij =S′ ij1 ∧S′ ij2 ∧L∧S′ ijn where S′ mij (j=1, L, n) represents the motion space of the j-th coupled branch coupled to the i-th basic branch, S′ ijk (k=1, L, n) represents the kinematic spiral of the kth kinematic pair on the jth closed-loop branch coupled to the i-th basic branch.

4. A multi-loop coupling mechanism degree of freedom calculation system based on geometric algebra, characterized by: include: Module M1: Determine the basic branch of the Twin-Bennett mechanism and obtain the initial motion space of the basic branch; Module M2: Find all coupled branches that form a closed loop with the base branch and calculate the branch motion space of the coupled branches; Module M3: Determine whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop among all the basic branches containing the closed loop; If yes, the motion space of the basic branch in the closed loop is equivalent to the motion space of the closed loop; If it does not belong to the category, the intersection of the motion space of the basic branch and the coupled branch in the closed loop is calculated to be equivalent to the motion space of the new basic branch; Module M4: Intersect the motion spaces of all branches of the equivalent mechanism to obtain the motion space of the output platform, thereby obtaining the degrees of freedom of the Twin-Bennett mechanism; In the module M3: Determine whether the motion space of the coupled branch in the closed loop belongs to the motion space of the basic branch in the closed loop. The determination method is: perform an outer product operation on the motion space of each branch in the closed loop, and determine whether the motion space of one branch belongs to the motion space of another branch based on the operation result; If the result of the outer product operation is 0, it means that the motion space of the coupled branch chain coupled with the basic branch in the closed loop belongs to the motion space of the basic branch. The coupled branch chain in the closed loop has no effect on the motion of the basic branch chain. Then the motion space S of the basic branch chain in the closed loop is converted to mi The motion space is equivalent to a closed loop; If the result of the outer product operation is not 0, the motion space S of the basic branch mi and the motion space S′ of the coupled branches coupled to the base branch mi1 , S′ mi2 , L, S′ mij Find the intersection and use this intersection to replace the motion space of the closed-loop output; In the module M4: The method for finding the intersection is: when the union of the allowable subspaces at the ends of the two branch kinematic chains is I6, the intersection S mM for: Where I6 represents the maximum order piece product in 6-dimensional geometric algebra, which is composed of the outer product of 6 unit vectors, I6=e1∧e2∧L∧e6, Yes, I6 inverse, When the union of the permissible subspaces at the ends of two branch kinematic chains is not equal to I6, the union of the two, I6, is used. u Instead of I6, the intersection S mM for:

5. The multi-loop coupling mechanism degree of freedom calculation system based on geometric algebra according to claim 4, characterized in that: In the module M1: Select the fixed platform and the moving platform of the mechanism, divide the basic branches, write the motion spiral of the kinematic pair on each basic branch, calculate the branch motion space of each basic branch, and obtain the initial motion space of the basic branch chain; The spiral is represented in six-dimensional geometric algebra G6 as a one-dimensional piece product: S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6 where e1, e2, L, e6 represent the basis in G6, and the scalar coefficient v i and b i , i = 1, 2, 3, represents the Plücker coordinates of the secondary axis of motion; The slice product S formed by the outer product operation of all n motion spirals on the i-th branch mi Represents the motion space of this branch: S mi =S i1 ∧S i2 ∧L∧S in Among them S ij (j=1, L, n) represents the motion spiral of the j-th motion pair on the i-th branch, and ∧ represents the outer product operation symbol.

6. The multi-loop coupling mechanism degree of freedom calculation system based on geometric algebra according to claim 4, characterized in that: In the module M2: Find all coupled branches that form a closed loop with the base branch, write the kinematic spiral of the kinematic pair on each coupled branch, and calculate the branch motion space of the coupled branch; The branch motion space S′ of the coupled branch chain coupled with the i-th basic branch in the closed loop mij Expressed as: S′ mij =S′ ij1 ∧S′ ij2 ∧L∧S′ ijn where S′ mij (j=1, L, n) represents the motion space of the j-th coupled branch coupled to the i-th basic branch, S′ ijk (k=1, L, n) represents the kinematic spiral of the kth kinematic pair on the jth closed-loop branch coupled to the i-th basic branch.

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