Singularity analysis method and system of multi-ring coupling mechanism based on geometric algebra

Abstract

The present invention provides a method and system for singularity analysis of a multi-loop coupling mechanism based on geometric algebra, comprising: selecting a fixed platform and a movable platform of the mechanism, dividing a basic branch and a coupled branch coupled thereto in a closed loop, writing the motion spiral of the kinematic pair on each branch, and obtaining its constraint space by calculating the motion space of each branch; determining the relationship between the constraint space of the coupled branch and the basic branch, and equivalently forming a new constraint space; performing the above calculation steps on all basic branches containing a closed loop; removing redundant constraint spirals on each new branch after the closed loop is equivalent, calculating the outer product of the constraint space of all equivalent branches, obtaining the outer product coefficient, and setting the outer product coefficient to 0 to obtain the singular configuration of the mechanism. The singularity analysis method of the present invention only needs to determine the motion spiral of each branch of the multi-loop coupling mechanism to calculate the branch constraint space, and then obtain the singular configuration of the mechanism, thereby conveniently and concisely performing singularity analysis on the multi-loop coupling mechanism.

Description

Technical Field

[0001] The present invention relates to the technical field of singular analysis of multi-ring coupling mechanisms, specifically, to a method and system for singular analysis of multi-ring coupling mechanisms based on geometric algebra, and especially to a method and system for singular analysis of multi-ring coupling mechanisms based on equivalent branches of geometric algebra. Background Art

[0002] Parallel mechanisms are multi-loop closed-chain mechanisms. Singularities are a key mechanical characteristic of parallel mechanisms. When a singularity occurs, the kinematic constraints of the mechanism become linearly dependent and invalid in the singular configuration, causing transient changes in the mechanism's degrees of freedom. When the mechanism is in a singular configuration, or in the vicinity of a singular configuration, its transfer performance deteriorates, and the mechanism's dynamic platform becomes uncontrollable. Singular configurations should be avoided when designing and applying parallel robots. Therefore, analyzing the singular configuration of a mechanism is crucial for its design, trajectory planning, and drive control.

[0003] Methods for singularity analysis of mechanisms can be broadly categorized into two types: algebraic and geometric. The geometric method primarily analyzes the rank of the Jacobian matrix of a parallel mechanism to identify singularities. The Jacobian matrix represents the mapping between the velocities of the actuating joints and the velocities of the end effectors. Singularities occur when the determinant of the Jacobian matrix is ​​zero. Gosselin and Angelss split the Jacobian matrix of a parallel robot into two parts related to the direct and inverse kinematic solutions. Huang Zhen et al. used the spiral method to establish the Jacobian matrix and analyzed the linear correlation between the force transmission spiral and the constraint force spiral to identify singularities. The algebraic method is a classic and traditional approach, but for complex parallel mechanisms, the determinant of the Jacobian matrix is ​​a complex nonlinear equation. Obtaining a symbolic solution that satisfies zero is extremely difficult, and it is also difficult to express the geometric meaning it represents. The essence of the geometric method is to identify singularities by analyzing the linear correlation between the motion vectors and constraint vectors of the moving joints in the parallel mechanism. Hunt used spiral theory to study singularities in parallel mechanisms. Merlet proposed a method based on Grassmann line geometry to analyze singularities. Amine also applied the Grassmann-Cayley algebra method to analyze singularities. Geometric methods can intuitively represent the geometric conditions of mechanism singularities, but they are also very difficult to apply to complex multi-loop coupled mechanisms.

[0004] In recent years, the structure of mechanisms has begun to develop towards more complex spatial structures. Parallel mechanisms have gradually evolved from simple open-loop branched chains to multi-loop coupled mechanisms. Multi-loop coupled mechanisms not only have the advantages of parallel mechanisms such as strong load-bearing capacity and high precision, but also have the advantages of series mechanisms such as large working space and flexible control. Multi-loop coupled mechanisms include complex, coupled, and closed sub-chains, resulting in complex kinematic relationships. For example, some folding and unfolding mechanisms and magic cube mechanisms have complex analysis processes and tedious calculations. Therefore, it is very necessary to propose an efficient and concise method for calculating the singularity analysis of multi-loop coupled mechanisms.

[0005] Therefore, it is necessary to propose a new technical solution to improve the above technical problems. Summary of the Invention

[0006] In view of the defects in the prior art, the purpose of the present invention is to provide a method and system for singularity analysis of multi-loop coupling mechanisms based on geometric algebra.

[0007] According to the present invention, a method for singularity analysis of a multi-ring coupling mechanism based on geometric algebra is provided, the method comprising the following steps:

[0008] 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 calculate the constraint space of each basic branch;

[0009] Step S2: Find all coupled branches that form a closed loop with the base branch, write out the motion spiral of the kinematic pair on each coupled branch, calculate the branch motion space on the coupled branch, and calculate the constraint space of the coupled branch;

[0010] Step S3: determining whether the constraint space of the coupled branch in the closed loop belongs to the constraint space of the basic branch in the closed loop, by performing an outer product operation on the constraint space of the coupled branch in the closed loop and the constraint space of the basic branch in the closed loop;

[0011] Step S4: If the outer product calculation result in step S3 is 0, it means that the constraint spaces of the two branches in the closed loop are linearly correlated, and the constraint space of the base branch in the closed loop is equivalent to the constraint space of the closed loop; if the outer product calculation result in step S3 is not 0, the union of the constraint spaces of the base branch and the coupled branch is equivalent to the constraint space of the closed loop output;

[0012] Step S5: perform the above calculations on all basic branches containing closed loops;

[0013] Step S6: Remove the redundant and identical constraint spirals on each new equivalent branch after the closed loop is equivalent, calculate the outer product of the constraint space of all equivalent branches, obtain the outer product coefficient, and set the outer product coefficient to 0 to obtain the singular configuration of the mechanism.

[0014] Preferably, in step S1 and step S2, the spiral is represented as a one-dimensional piece product in six-dimensional geometric algebra G6:

[0015] S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6

[0016] 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.

[0017] Preferably, the outer product operation of all n motion spirals on the i-th branch forms the slice product S mi Represents the motion space of this branch:

[0018] S mi =S i1 ∧S i2 ∧…∧S in

[0019] 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 operator symbol;

[0020] The branch constraint space imposed by the i-th branch on the moving platform is denoted as S ci :

[0021]

[0022] in represents the inverse of the unit pseudoscalar in G6, and "Δ" represents the direct transformation operator, which operates by exchanging the principal coefficients v of the spiral. i and the secondary coefficient b i .

[0023] Preferably, in step S3 and step S4, the method for processing the equivalent branched chains is:

[0024] After determining the basic branch, find the coupling branch in the closed loop that is coupled with the basic branch. The constraint space of the i-th basic branch is S ci , the coupled branch constraint space coupled with the i-th basic branch is S ci1 ,S ci2 ,…,S cij , calculate the outer product of the constraint space of each branch on the closed loop, expressed as:

[0025] S ci ∧S cij

[0026] Preferably, if the outer product result is 0, it means that the constraint spaces of the two branches are linearly correlated, and one of the branches is used to equivalently replace the closed-loop structure for singularity analysis;

[0027] If the outer product result is not 0, decompose the constraint space into constraint subspaces, determine and remove redundant constraint subspaces, and make a union of linearly unrelated constraint subspaces to replace the closed-loop constraint space.

[0028] The present invention also provides a multi-ring coupling mechanism singularity analysis system based on geometric algebra, the system comprising the following modules:

[0029] 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 calculate the constraint space of each basic branch;

[0030] Module M2: Find all coupled branches that form a closed loop with the base branch, write the kinematic spiral of each coupled branch, calculate the branch motion space on the coupled branch, and calculate the constraint space of the coupled branch;

[0031] Module M3: Determine whether the constraint space of the coupled branch in the closed loop belongs to the constraint space of the basic branch in the closed loop. The determination method is: calculate the outer product operation of the constraint space of the coupled branch in the closed loop and the constraint space of the basic branch in the closed loop;

[0032] Module M4: If the result of the outer product operation in module M3 is 0, it means that the constraint spaces of the two branches in the closed loop are linearly related, and the constraint space of the basic branch in the closed loop is equivalent to the constraint space of the closed loop; if the result of the outer product operation in module M3 is not 0, the union of the constraint spaces of the basic branch and the coupled branch is equivalent to the constraint space of the closed loop output;

[0033] Module M5: Perform the above module calculations on all basic branches containing closed loops;

[0034] Module M6: Remove the redundant and identical constraint spirals on each new equivalent branch after the closed loop is equivalent, calculate the outer product of the constraint space of all equivalent branches, obtain the outer product coefficient, and set the outer product coefficient to 0 to obtain the singular configuration of the mechanism.

[0035] Preferably, in the modules M1 and M2, the spiral is represented as a one-dimensional patch product in six-dimensional geometric algebra G6:

[0036] S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6

[0037] 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.

[0038] Preferably, the outer product operation of all n motion spirals on the i-th branch forms the slice product S mi Represents the motion space of this branch:

[0039] S mi =S i1 ∧S i2 ∧…∧S in

[0040] 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 operator symbol;

[0041] The branch constraint space imposed by the i-th branch on the moving platform is denoted as S ci :

[0042]

[0043] in represents the inverse of the unit pseudoscalar in G6, and "Δ" represents the direct transformation operator, which operates by exchanging the principal coefficients v of the spiral. i and the secondary coefficient b i .

[0044] Preferably, in the modules M3 and M4, the equivalent branch processing system is:

[0045] After determining the basic branch, find the coupling branch in the closed loop that is coupled with the basic branch. The constraint space of the i-th basic branch is S ci , the coupled branch constraint space coupled with the i-th basic branch is S ci1 ,S ci2 ,…,S cij , calculate the outer product of the constraint space of each branch on the closed loop, expressed as:

[0046] S ci ∧S cij

[0047] Preferably, if the outer product result is 0, it means that the constraint spaces of the two branches are linearly correlated, and one of the branches is used to equivalently replace the closed-loop structure for singularity analysis;

[0048] If the outer product result is not 0, decompose the constraint space into constraint subspaces, determine and remove redundant constraint subspaces, and make a union of linearly unrelated constraint subspaces to replace the closed-loop constraint space.

[0049] Compared with the prior art, the present invention has the following beneficial effects:

[0050] 1. The calculation process of the present invention is simple. The singularity analysis method of the present invention only needs to determine the motion spiral of the closed-loop branch of the multi-loop coupling mechanism to calculate the branch constraint space, and the operation is simple;

[0051] 2. The physical meaning of the present invention is clear. The present invention uses geometric algebra to solve the constraint space, and equates the constraint space of the closed loop of the multi-loop coupling mechanism to the constraint space of the new basic branch. This is equivalent to decoupling the multi-loop coupling mechanism and converting it into a parallel mechanism with open-chain branches. The analysis process is simple. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] 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:

[0053] Figure 1 Flowchart of the equivalent branched chain singular analysis method of the present invention;

[0054] Figure 2 A schematic diagram of a Bennett mechanism according to an embodiment of the present invention;

[0055] Figure 3 A schematic diagram of a Twin-Bennett mechanism according to an embodiment of the present invention;

[0056] Figure 4 A Twin-Bennett mechanism diagram of an embodiment of the present invention;

[0057] Figure 5 A position diagram of a joint coordinate system according to an embodiment of the present invention;

[0058] Figure 6 2 is a motion spiral diagram of the Twin-Bennett mechanism according to an embodiment of the present invention. DETAILED DESCRIPTION

[0059] 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.

[0060] Example 1:

[0061] According to the present invention, a method for singularity analysis of a multi-ring coupling mechanism based on geometric algebra is provided, the method comprising the following steps:

[0062] 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 calculate the constraint space of each basic branch;

[0063] Step S2: Find all coupled branches that form a closed loop with the base branch, write out the motion spiral of the kinematic pair on each coupled branch, calculate the branch motion space on the coupled branch, and calculate the constraint space of the coupled branch;

[0064] In steps S1 and S2, the spiral is represented as a one-dimensional piece product in six-dimensional geometric algebra G6:

[0065] S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6

[0066] 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.

[0067] 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:

[0068] S mi =S i1 ∧S i2 ∧…∧S in

[0069] 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 operator symbol;

[0070] The branch constraint space imposed by the i-th branch on the moving platform is denoted as S ci :

[0071]

[0072] in represents the inverse of the unit pseudoscalar in G6, and "Δ" represents the direct transformation operator, which operates by exchanging the principal coefficients v of the spiral. i and the secondary coefficient b i .

[0073] Step S3: determining whether the constraint space of the coupled branch in the closed loop belongs to the constraint space of the basic branch in the closed loop, by performing an outer product operation on the constraint space of the coupled branch in the closed loop and the constraint space of the basic branch in the closed loop;

[0074] Step S4: If the outer product calculation result in step S3 is 0, it means that the constraint spaces of the two branches in the closed loop are linearly correlated, and the constraint space of the base branch in the closed loop is equivalent to the constraint space of the closed loop; if the outer product calculation result in step S3 is not 0, the union of the constraint spaces of the base branch and the coupled branch is equivalent to the constraint space of the closed loop output;

[0075] In step S3 and step S4, the processing method of the equivalent branch chain is:

[0076] After determining the basic branch, find the coupling branch in the closed loop that is coupled with the basic branch. The constraint space of the i-th basic branch is S ci , the coupled branch constraint space coupled with the i-th basic branch is S ci1 ,S ci2 ,…,S cij , calculate the outer product of the constraint space of each branch on the closed loop, expressed as:

[0077] S ci ∧S cij

[0078] If the outer product result is 0, it means that the constraint spaces of the two branches are linearly related, and one of the branches is used to replace the closed loop structure for singular analysis;

[0079] If the outer product result is not 0, decompose the constraint space into constraint subspaces, determine and remove redundant constraint subspaces, and make a union of linearly unrelated constraint subspaces to replace the closed-loop constraint space.

[0080] Step S5: perform the above calculations on all basic branches containing closed loops;

[0081] Step S6: Remove the redundant and identical constraint spirals on each new equivalent branch after the closed loop is equivalent, calculate the outer product of the constraint space of all equivalent branches, obtain the outer product coefficient, and set the outer product coefficient to 0 to obtain the singular configuration of the mechanism.

[0082] Example 2:

[0083] Example 2 is a preferred example of Example 1 and is used to illustrate the present invention in more detail.

[0084] The present invention also provides a multi-ring coupling mechanism singularity analysis system based on geometric algebra, which includes the following modules:

[0085] 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 calculate the constraint space of each basic branch;

[0086] Module M2: Find all coupled branches that form a closed loop with the base branch, write the kinematic spiral of each coupled branch, calculate the branch motion space on the coupled branch, and calculate the constraint space of the coupled branch;

[0087] In modules M1 and M2, the spiral is represented as a one-dimensional piece product in six-dimensional geometric algebra G6:

[0088] S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6

[0089] 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.

[0090] 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:

[0091] S mi =S i1 ∧S i2 ∧…∧S in

[0092] 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 operator symbol;

[0093] The branch constraint space imposed by the i-th branch on the moving platform is denoted as S ci :

[0094]

[0095] in represents the inverse of the unit pseudoscalar in G6, and "Δ" represents the direct transformation operator, which operates by exchanging the principal coefficients v of the spiral. i and the secondary coefficient b i .

[0096] Module M3: Determine whether the constraint space of the coupled branch in the closed loop belongs to the constraint space of the basic branch in the closed loop. The determination method is: calculate the outer product operation of the constraint space of the coupled branch in the closed loop and the constraint space of the basic branch in the closed loop;

[0097] Module M4: If the result of the outer product operation in module M3 is 0, it means that the constraint spaces of the two branches in the closed loop are linearly related, and the constraint space of the basic branch in the closed loop is equivalent to the constraint space of the closed loop; if the result of the outer product operation in module M3 is not 0, the union of the constraint spaces of the basic branch and the coupled branch is equivalent to the constraint space of the closed loop output;

[0098] In modules M3 and M4, the equivalent branch chain processing system is:

[0099] After determining the basic branch, find the coupling branch in the closed loop that is coupled with the basic branch. The constraint space of the i-th basic branch is S ci , the coupled branch constraint space coupled with the i-th basic branch is S ci1 ,S ci2 ,…,S cij , calculate the outer product of the constraint space of each branch on the closed loop, expressed as:

[0100] S ci ∧S cij

[0101] If the outer product result is 0, it means that the constraint spaces of the two branches are linearly related, and one of the branches is used to replace the closed loop structure for singular analysis;

[0102] If the outer product result is not 0, decompose the constraint space into constraint subspaces, determine and remove redundant constraint subspaces, and make a union of linearly unrelated constraint subspaces to replace the closed-loop constraint space.

[0103] Module M5: Perform the above module calculations on all basic branches containing closed loops;

[0104] Module M6: Remove the redundant and identical constraint spirals on each new equivalent branch after the closed loop is equivalent, calculate the outer product of the constraint space of all equivalent branches, obtain the outer product coefficient, and set the outer product coefficient to 0 to obtain the singular configuration of the mechanism.

[0105] Example 3:

[0106] Example 3 is a preferred example of Example 1 and is used to illustrate the present invention in more detail.

[0107] The present invention proposes a singularity analysis method for a multi-loop coupling mechanism based on equivalent branches using the mathematical tool of geometric algebra. The method has the characteristics of a simple analysis process and a convenient calculation process.

[0108] The present invention provides a method for singularity analysis of a multi-ring coupling mechanism based on equivalent branches of geometric algebra, which is performed in the following steps:

[0109] Step 1: 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, obtain the branch motion space of each basic branch, and then obtain the constraint space of each basic branch;

[0110] The spiral is represented in six-dimensional geometric algebra G6 as a one-dimensional piece product:

[0111] S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6

[0112] 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 axes of motion.

[0113] 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:

[0114] S mi =S i1 ∧S i2 ∧…∧S in

[0115] 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.

[0116] The branch constraint space imposed by the i-th branch on the moving platform can be recorded as S ci , the motion space S mi It is obtained by the joint operation of duality and direct transformation, which is expressed as:

[0117]

[0118] Indicates that the motion space S mi Perform dual operation, where represents the inverse of the unit pseudoscalar in G6, and "Δ" represents the direct transformation operator, which operates by exchanging the principal coefficients v of the spiral. i and the secondary coefficient b i ,The physical meaning of the spiral is expressed as the ,element that constitutes the motion space and the force space, and ,the transformation of the two spaces can be realized by the ,direction transformation Δ.

[0119] Step 2: Find all coupled branches coupled with the base branch in the closed loop, write the kinematic spiral of the kinematic pair on each coupled branch, calculate the branch motion space on the coupled branch, and calculate the constraint space on the coupled branch;

[0120] The branch motion space S′ of the coupled branch coupled to the i-th basic branch in the closed loop mij Expressed as:

[0121] S′ mij =S′ ij1 ∧S′ ij2 ∧…∧S′ ijn

[0122] where S′ mij (j=1,…,n) represents the motion space of the j-th coupled branch coupled to the i-th basic branch, S′ ijk (k=1,…,n) represents the kinematic spiral of the kth kinematic pair on the jth closed-loop branch coupled to the i-th basic branch;

[0123] Step 3: Determine whether the constraint space of the coupled branch chain coupled with the base branch in the closed loop belongs to the constraint space of the base branch in the closed loop. The determination method is to calculate the outer product of the constraint space of the coupled branch chain coupled with the base branch in the closed loop and the constraint space of the base branch in the closed loop, and discuss the outer product result.

[0124] After determining the basic branch, find the coupled branches coupled with the basic branch. The constraint space of the i-th basic branch is S ci , the coupled branch constraint space coupled with the i-th basic branch is S ci1 ,S ci2 ,…,S cij , calculate the outer product of the constraint space of the basic branch and the coupled branch on the closed loop, expressed as:

[0125] S ci ∧S cij

[0126] Step 4: If the outer product calculation result in step 3 is 0, it means that the constraint spaces of the two branches in the closed loop are linearly related. In this case, the constraint space of the base branch in the closed loop is equivalent to the constraint space of the closed loop. If the outer product calculation result in step 3 is not 0, the union of the constraint spaces of the base branch and the coupled branch is equivalent to the constraint space of the closed loop output.

[0127] The specific processing method of equivalent branch chain is:

[0128] If the outer product result is 0, it means that the constraint spaces of the two branches are linearly related, then the constraint space of the available basic branch is S ci Equivalently replace the constraint space of the closed loop structure for singular analysis;

[0129] If the outer product result is not 0, the constraint space S of the basic branch ci and the constraint space S of the coupled branches coupled to the base branch ci1 ,S ci2 ,…,Scij Find the union and use this union to replace the constraint space of the closed-loop output.

[0130] The method for finding the union is: ci and S cij Decompose it into constraint subspaces consisting only of 1st-order slice products, determine and remove redundant constraint subspaces, and make a union of linearly unrelated constraint subspaces to replace the closed-loop constraint space.

[0131] Step 5: Perform the above calculations on all basic branches containing closed loops;

[0132] Step 6: Remove the redundant and identical constraint spirals on each new equivalent branch after the closed loop is equivalent, calculate the outer product of the constraint space of all equivalent branches, obtain the outer product coefficient, and set the outer product coefficient to 0 to obtain the singular configuration of the mechanism.

[0133] The present invention proposes a method for equivalent branch chain singularity analysis. This method calculates the constraint space of each branch on a closed loop based on geometric algebra, calculates the outer product between the constraint spaces of each branch on the closed loop, and determines the linear correlation between the branch constraint spaces. If the outer product is 0, indicating that the constraint spaces of the two branches are linearly correlated, one of the branches can be used to equivalently replace the constraint space of the closed loop structure for singularity analysis. If the outer product is not 0, the constraint space is decomposed into constraint subspaces, and redundant constraint spaces are identified and removed. The linearly uncorrelated constraint subspaces are then unioned to replace the constraint space of the closed loop for singularity analysis. Finally, the constraint spaces of all branches after the closed loop is equivalent are unioned to obtain the outer product coefficient, which is then set to 0 to obtain the singular configuration of the mechanism.

[0134] Singularity analysis of Twin-Bennett closed-loop mechanism:

[0135] Bennett mechanism is the only single-degree-of-freedom spatial mechanism composed of four revolute pairs. It plays an important role in mechanism theory and is recognized as the most classic over-constrained mechanism with the least number of rods and the highest degree of constraint. In addition, many mechanisms are synthesized based on Bennett mechanism as a unit. Bennett mechanism is a single-degree-of-freedom spatial single-loop mechanism ( Figure 2 (Figure 1 is a simplified diagram of the mechanism) and consists of four connecting rods and four revolute pairs. The center points of the four revolute pairs are A, B, C, and D. The connecting rods in relative positions have the same properties. The lengths of the four connecting rods are AB = CD = m and AD = BA = n, respectively. The torsion angles of the revolute pairs at the ends of rods AB, CD, AD, and BC are denoted by α. AB ,α DC ,α AD and α BC , and α AB =α DC=α,α AD =α BC =β. The connecting rod length and the torsion angle between the two ends of the connecting rod's rotating secondary axis satisfy the following relationship:

[0136]

[0137] 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 3 (Figure 2 is a simplified diagram of the mechanism), which contains 6 revolute pairs and 7 connecting rods. The Twin-Bennett mechanism satisfies the following geometric relationships:

[0138] OA=CB=CD=a

[0139] OC=AB=AD=b

[0140] α OA =α CB =α1=-α,α CD =α2=α

[0141] α OC =α AB =α AD =β

[0142]

[0143]

[0144]

[0145] 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.

[0146] The Twin-Bennett mechanism uses rod e as the moving platform and revolving pair 6 as the driving pair. The branch from rod e to the frame base is:

[0147]

[0148] The branched chain 1-a-2-b-3 is coupled with the base branch 6-d-1 to form a closed loop.

[0149] The motion spirals of the 6 revolving pairs of the mechanism are:

[0150]

[0151] The motion space S of the basic branch 6-d-1 m1 Expanded from the upper spirals S1 and S6:

[0152] S m1 =S1∧S6

[0153] =-s(β)s(θ)e 1,3 +s(β)c(θ)e 2,3 -mc(β)s(θ)e 3,4 +mc(β)c(θ)e 3,5 -ms(β)e 3,6

[0154] Constraint space S of basic branch 6-d-1 c1 can be written as:

[0155]

[0156] The motion space S of the basic branch 4-f-5 m2 Expanded from the spirals S4 and S5 above:

[0157] S m2 =S4∧S5

[0158] =s(β)s(θ)s(α)e 1,2,3 +s(β)s(θ)lc(α)e 1,3,5 -s(β)s(θ)ls(α)e 1,3,6 -mc(β)s(θ)s(α)e 2,3,4 +(mc(β)c(θ)s(α)-s(β)c(θ)lc(α))e 2,3,5 +(s(β)c(θ)ls(α)-ms(β)s(α))e 2,3,6 +mc(β)s(θ)lc(α)e 3,4,5 -mc(β)s(θ)ls(α)e 3,4,6 +(mc(β)c(θ)ls(α)-ms(β)lc(α))e 3,5,6

[0159] Constraint space S of basic branch 4-f-5 c2 can be written as:

[0160]

[0161] The motion space S' of the branch 1-a-2-b-3 coupled with the base branch 6-d-1 m11Expanded from the spirals S1, S2, and S3 above:

[0162] S' m11 =S1∧S2∧S3

[0163] =s(β)s(θ)s(α)e 1,2,3 -s(β)s(θ)lc(α)e 1,3,5 -s(β)s(θ)ls(α)e 1,3,6 -mc(β)s(θ)s(α)e 2,3,4 +(mc(β)c(θ)s(α)+s(β)c(θ)lc(α))e 2,3,5 +(s(β)c(θ)ls(α)-ms(β)s(α))e 2,3,6 -mc(β)s(θ)lc(α)e 3,4,5 -mc(β)s(θ)ls(α)e 3,4,6 +(mc(β)c(θ)ls(α)+ms(β)lc(α))e 3,5,6

[0164] Constraint space S of closed-loop branch 1-a-2-b-3 c11 can be written as:

[0165]

[0166] Equivalence of closed-loop branched chain:

[0167] Perform piecewise decomposition on the coupled branch 1-a-2-b-3 constraint space and decompose it into three first-order piecewise products to obtain its three constraint subspaces b1, b2, and b3:

[0168] S c11 =b1∧b2∧b3

[0169]

[0170]

[0171]

[0172] The three first-order slice products b1, b2, and b3 are respectively combined with the constraint space S of the basic branch 6-d-1 c1 Perform outer product operation:

[0173] S c1 ∧b1=0

[0174] S c1 ∧b2=0

[0175] S c1 ∧b3=0

[0176] After calculation, the outer products of the constraint space of chain 6-d-1 and the three constraint subspaces of chain 3-b-2-a-1 are all 0. According to the operation rules of geometric algebra, it shows that the constraint spaces of the two branches are linearly related. It further shows that the constraint spaces of the two branches can be replaced by each other equivalently. Therefore, for this mechanism, the constraint space of the closed loop formed by the basic branch 6-d-1 and the coupling branch 1-a-2-b-3 is calculated equivalently using the constraint space of the basic branch 6-d-1.

[0177] Therefore, the constraint space of a closed loop can be equivalent to the constraint space of chain 6-d-1. Let the revolute joint 6 be the driving joint and lock the driving joint S6. The constraint space of the motion space of the basic branch 6-d-1 can be written as:

[0178] S m1 =S1

[0179]

[0180] After determining and removing redundant constraints, in order to analyze the linear correlation of the constraint spaces of the two branches, we perform outer product operations on them and obtain:

[0181]

[0182] The polynomial Q includes the structural parameters of the mechanism. When the structural parameters of the mechanism are given, the mechanism is in a specific posture state and Q = 0. When the drive angle θ is 0° or 180°, the outer product coefficient is 0, and the mechanism will be in a singular position.

[0183] Singular position verification:

[0184] The Twin-Bennett mechanism is composed of two Bennett mechanisms, which are arranged symmetrically on both sides. The motion conditions on both sides are the same as those of the Bennett mechanism. The input angle θ is the only variable. The motion spiral on the moving platform under different configurations is as follows: Figure 6 As shown, line ① represents the direction of the permissible spiral on the moving platform under different configurations, and line ② represents the trace of a point on the permissible spiral. When the angle θ is 0° and 180°, the direction of the moving spiral changes suddenly, which is the singular position of the mechanism, verifying the correctness of the singularity analysis method of the present invention.

[0185] Those skilled in the art may understand this embodiment as a more specific description of Embodiment 1 and Embodiment 2.

[0186] Those skilled in the art will appreciate that, in addition to implementing the system and its various devices, modules, and units provided by the present invention in purely computer-readable program code, it is entirely possible to implement the same functions of the system and its various devices, modules, and units provided by the present invention in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers by logically programming the method steps. Therefore, the system and its various devices, modules, and units provided by the present invention can be considered a hardware component, and the devices, modules, and units included therein for implementing various functions can also be considered as structures within the hardware component; the devices, modules, and units for implementing various functions can also be considered as both software modules implementing the method and structures within the hardware component.

[0187] 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 singularity analysis of multi-ring coupling mechanisms based on geometric algebra, characterized by: The method comprises the following steps: 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 calculate the constraint space of each basic branch; Step S2: Find all coupled branches that form a closed loop with the base branch, write out the motion spiral of the kinematic pair on each coupled branch, calculate the branch motion space on the coupled branch, and calculate the constraint space of the coupled branch; Step S3: determining whether the constraint space of the coupled branch in the closed loop belongs to the constraint space of the basic branch in the closed loop, by performing an outer product operation on the constraint space of the coupled branch in the closed loop and the constraint space of the basic branch in the closed loop; Step S4: If the outer product calculation result in step S3 is 0, it means that the constraint spaces of the two branches in the closed loop are linearly correlated, and the constraint space of the base branch in the closed loop is equivalent to the constraint space of the closed loop; if the outer product calculation result in step S3 is not 0, the union of the constraint spaces of the base branch and the coupled branch is equivalent to the constraint space of the closed loop output; Step S5: perform the above calculations on all basic branches containing closed loops; Step S6: Remove the redundant and identical constraint spirals on each new equivalent branch after the closed loop is equivalent, calculate the outer product of the constraint space of all equivalent branches, obtain the outer product coefficient, and set the outer product coefficient to 0 to obtain the singular configuration of the mechanism.

2. The method for singularity analysis of multi-ring coupling mechanisms based on geometric algebra according to claim 1, characterized in that: In steps S1 and S2, the spiral is represented as a one-dimensional piece product in six-dimensional geometric algebra G6: S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6 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.

3. The method for singularity analysis of multi-ring coupling mechanisms based on geometric algebra according to claim 2, characterized in that: 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 ∧…∧S in 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 operator symbol; The branch constraint space imposed by the i-th branch on the moving platform is denoted as S ci : in represents the inverse of the unit pseudoscalar in G6, "Δ" represents the direct transformation operator, which operates by exchanging the principal coefficients v of the spiral. i and the secondary coefficient b i .

4. The method for singularity analysis of multi-ring coupling mechanisms based on geometric algebra according to claim 1, characterized in that: In step S3 and step S4, the method for processing the equivalent branch chain is: After determining the basic branch, find the coupling branch in the closed loop that is coupled with the basic branch. The constraint space of the i-th basic branch is S ci , the coupled branch constraint space coupled with the i-th basic branch is S ci1 ,S ci2 ,…,S cij , calculate the outer product of the constraint space of each branch on the closed loop, expressed as: S ci ∧S cij。 5. The method for singularity analysis of multi-ring coupling mechanisms based on geometric algebra according to claim 4, characterized in that: If the outer product result is 0, it means that the constraint spaces of the two branches are linearly related, and one of the branches is used to replace the closed loop structure for singular analysis; If the outer product result is not 0, decompose the constraint space into constraint subspaces, determine and remove redundant constraint subspaces, and make a union of linearly unrelated constraint subspaces to replace the closed-loop constraint space.

6. A multi-ring coupling mechanism singularity analysis system based on geometric algebra, characterized by: The system includes the following modules: 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 calculate the constraint space of each basic branch; Module M2: Find all coupled branches that form a closed loop with the base branch, write the kinematic spiral of each coupled branch, calculate the branch motion space on the coupled branch, and calculate the constraint space of the coupled branch; Module M3: Determine whether the constraint space of the coupled branch in the closed loop belongs to the constraint space of the basic branch in the closed loop. The determination method is: calculate the outer product operation of the constraint space of the coupled branch in the closed loop and the constraint space of the basic branch in the closed loop; Module M4: If the result of the outer product operation in module M3 is 0, it means that the constraint spaces of the two branches in the closed loop are linearly related, and the constraint space of the basic branch in the closed loop is equivalent to the constraint space of the closed loop; if the result of the outer product operation in module M3 is not 0, the union of the constraint spaces of the basic branch and the coupled branch is equivalent to the constraint space of the closed loop output; Module M5: Perform the above module calculations on all basic branches containing closed loops; Module M6: Remove the redundant and identical constraint spirals on each new equivalent branch after the closed loop is equivalent, calculate the outer product of the constraint space of all equivalent branches, obtain the outer product coefficient, and set the outer product coefficient to 0 to obtain the singular configuration of the mechanism.

7. The multi-ring coupling mechanism singularity analysis system based on geometric algebra according to claim 6, characterized in that: In the modules M1 and M2, the spiral is represented as a one-dimensional piece product in six-dimensional geometric algebra G6: S=v1e1+v2e2+v3e3+b1e4+b2e5+b3e6 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.

8. The multi-ring coupling mechanism singularity analysis system based on geometric algebra according to claim 7, characterized in that: 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 ∧…∧S in 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 operator symbol; The branch constraint space imposed by the i-th branch on the moving platform is denoted as S ci : in represents the inverse of the unit pseudoscalar in G6, "Δ" represents the direct transformation operator, which operates by exchanging the principal coefficients v of the spiral. i and the secondary coefficient b i .

9. The multi-ring coupling mechanism singularity analysis system based on geometric algebra according to claim 6, characterized in that: In the modules M3 and M4, the equivalent branch chain processing system is: After determining the basic branch, find the coupling branch in the closed loop that is coupled with the basic branch. The constraint space of the i-th basic branch is S ci , the coupled branch constraint space coupled with the i-th basic branch is S ci1 ,S ci2 ,…,S cij , calculate the outer product of the constraint space of each branch on the closed loop, expressed as: S ci ∧S cij。 10. The multi-ring coupling mechanism singularity analysis system based on geometric algebra according to claim 9, characterized in that: If the outer product result is 0, it means that the constraint spaces of the two branches are linearly related, and one of the branches is used to replace the closed loop structure for singular analysis; If the outer product result is not 0, decompose the constraint space into constraint subspaces, determine and remove redundant constraint subspaces, and make a union of linearly unrelated constraint subspaces to replace the closed-loop constraint space.

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