CN102689306A - Screw space intersection-based parallel mechanism degree-of-freedom analysis method - Google Patents

Abstract

The invention discloses a screw space intersection-based parallel mechanism degree-of-freedom analysis method. The method comprises the following steps of: calculating the number of dimensions of a branch screw space; calculating degree of freedom of a double-chain single-ring mechanism moving platform; calculating an intersection space of screw spaces of two branches; calculating the degree of freedom of a multi-ring mechanism moving platform; arranging a driver of the parallel mechanism; recognizing the local degree of freedom which probably exists in the branches; and calculating the overall degree of freedom of the parallel mechanism. The degree of freedom of the moving platforms of the parallel mechanisms of all types and the overall degree of freedom of the mechanism can be correctly calculated without exception; for the single-ring mechanism, the intersection space of the double-branch screw space is not required to be solved, but the degree of freedom of the moving platform can be directly obtained; and for the condition that the number of dimensions of the union set of the screw spaces of the two branches is equal to the rank of a screw matrix of the branch or equal to the sum of ranks of screw matrixes of the two branches, the intersection space can be directly obtained without any operation.

Description

The parallel institution free degree based on the spiral space intersection operation is divided the folding method

Technical field

The present invention relates to a kind of parallel institution freedom analysis method, especially relate to a kind of parallel institution freedom analysis method based on the spiral space intersection operation.

Background technology

It is the basis of analyzing and design new machine that mechanism freedom calculates.Since the Chebychev-Gr of classics ü bler-Kutzbach (CGK) formula proposes, the existing about 150 years history of mechanism freedom research.In very long production practices, the CGK formula can correctly calculate the free degree of most of mechanisms easily.Yet people also find minority mechanism especially the free degree of space parallel mechanism can not correctly draw with the CGK formula.So many scholars further investigate mechanism freedom, and the new method that some calculate the frees degree has been proposed.The new method of research institution's free degree, one side can correctly and unlimitedly be calculated the free degree of all types mechanism; The essence of mechanism kinematic can more profoundly be familiar with in the aspect in addition, can form the effective ways of synthesis of mechanism.

(Higher Education Publishing House, Beijing proposed the CGK freedom calculation formula of correction in 1997:18-28) to Huang Zhen at monograph " parallel robot mechanism scientific principle touch upon control " in 1997.Based on the rank of constraint screw theory definition mechanism, obtain the exponent number of mechanism through finding the solution general constraint, thus the CGK formula that obtains revising.Zhao Jingshan in 2004 " Mechanism and Machine Theory " (2004, in paper 39:621-643) " A theory of degrees of freedom for mechanisms " freedom analysis method based on the reciprocity spiral has been proposed.Find the solution side chain constraint spiral by the kinematic screw of each bar side chain of parallel institution, find the solution the reciprocity spiral of all constraint spirals again, thereby obtain the motion of mechanism spiral.These kinematic screw numbers are exactly the free degree of mechanism's moving platform, yet need not be equal to the whole free degree of mechanism.(2005, the linear transformation that has proposed to be used for the calculation mechanism free degree in paper 24:690-711) " Mobility and spatiality of parallel robots revisited via theory of linear transformations " was theoretical at " European Journal of Mechanics A – Solids " in 2005 for Gogu.Come the dimension of calculation mechanism operating space through the linear transformation of Jacobian matrix.This method need be calculated the common factor of each bar side chain operating space, thereby is applicable to that each bar side chain kinematic pair axis is parallel to each other or vertical mechanism; But for existing the kinematic pair axis not to be parallel to each other or vertical mechanism, the calculating of occuring simultaneously in the side chain operating space is difficulty very.Yu equal 2009 " Robotica " (2009, adopt in paper 27:915-927) " Mobility analysis of complex joints by means of screw theory " screw theory to analyze the secondary free degree of compound movement.The compound movement pair constitutes by some simple motions are secondary compound.Through the equivalence transformation CGK freedom calculation formula that obtain revise of kinematic pair spiral with the constraint spiral.

Summary of the invention

The object of the present invention is to provide a kind of parallel institution freedom analysis method based on the spiral space intersection operation.

The technical scheme that the present invention adopts is that the step of this method is following:

(1) dimension of calculating side chain spiral space: establish $ ⁱJ is the spiral vector of j kinematic pair in the i bar side chain, and then the set of the kinematic screw of each kinematic pair composition of side chain i is M _i={ $ ⁱ1, $ ⁱ2 ..., $ ⁱn _i, the rank of matrix that set constitutes is Rank (M _i)≤6, so the kinematic screw of side chain i constitutes Rank (M _i) the dimension spiral space, be designated as A _i, spiral space A _iDimension Dim (A _i)=Rank (M _i);

(2) calculate two side chain single loop mechanism moving platform frees degree: a side chain spiral space A ₁Dimension Dim (A ₁)=Rank (M ₁), another side chain spiral space A ₂Dimension Dim (A ₂)=Rank (M ₂), M ₁∪ M ₂={ $ ¹1, $ ¹2 ..., $ ¹N1, $ ²1, $ ²2 ..., $ ²n ₂, spiral space A then ₁And A ₂The dimension Dim (A in union space ₁∪ A ₂)=Rank (M ₁∪ M ₂), so monocycle parallel institution moving platform freedom calculation does

DOF(MovingPlatform)=Rank(M ₁)+Rank(M ₂)-Rank(M ₁∪M ₂) (1)；

(3) the common factor space of two side chain spiral spaces of calculating:

Kinematic screw set M according to two side chains<sub >1</sub>And M<sub >2</sub>And union M<sub >1</sub>∪ M<sub >2</sub>, calculate the spiral rank of matrix N that these set constitute<sub >1</sub>=Rank (M<sub >1</sub>), N<sub >2</sub>=Rank (M<sub >2</sub>), N<sub >3</sub>=Rank (M<sub >1</sub>∪ M<sub >2</sub>), suppose N<sub >1</sub>>=N<sub >2</sub>, obvious N<sub >3</sub>>=N<sub >1</sub>If, N<sub >3</sub>=N<sub >1</sub>, A then<sub >1</sub>∩ A<sub >2</sub>=A<sub >2</sub>, i.e. A<sub >2</sub>Be exactly spiral space A<sub >1</sub>∩ A<sub >2</sub>The common factor space, if N<sub >3</sub>=N<sub >1</sub>+ N<sub >2</sub>, M is described<sub >2</sub>In all spiral vectors to M<sub >1</sub>∪ M<sub >2</sub>The number of middle spiral vector all has contribution, i.e. spiral space A<sub >1</sub>With spiral space A<sub >2</sub>Do not comprise mutually, at this moment<img file="BDA00001742003200021.GIF" he="44" img-content="drawing" img-format="GIF" inline="yes" orientation="portrait" wi="294" />Be that mechanism can not move, if N<sub >1</sub><n<sub >3</sub><n<sub >1</sub>+ N<sub >2</sub>, M is described<sub >2</sub>Middle part screw amount is to M<sub >1</sub>∪ M<sub >2</sub>The number of middle spiral vector has contribution, A<sub >1</sub>∩ A<sub >2</sub>Basic spiral number in the spiral space is N<sub >1</sub>+ N<sub >2</sub>-N<sub >3</sub>, need calculate A this moment as follows<sub >1</sub>∩ A<sub >2</sub>Basic spiral:

(3.1) in 6 dimension spaces, calculate spiral space A respectively ₁And A ₂The supplementary set space, be designated as T ₁And T ₂, obvious T ₁∪ A ₁=T ₂∪ A ₂=R ₆, use as if spiral is vectorial (l, m, n, p, q, r) ^TExpression, then the spiral matrix that constituted of spiral set does

$S=\left[\begin{array}{cccc}{l}_1& {\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="120608142648.png"\; state="\{\{state\}\}">l</figure-callout>}}_2& \&CenterDot;\&CenterDot;\&CenterDot;& l_{n1}\\ m_1& {\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="120608142648.png"\; state="\{\{state\}\}">m</figure-callout>}}_2& \&CenterDot;\&CenterDot;\&CenterDot;& m_{n1}\\ n_1& {\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="120608142648.png"\; state="\{\{state\}\}">n</figure-callout>}}_2& \&CenterDot;\&CenterDot;\&CenterDot;& n_{n1}\\ p_1& {\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="120608142648.png"\; state="\{\{state\}\}">p</figure-callout>}}_2& \&CenterDot;\&CenterDot;\&CenterDot;& p_{n1}\\ q_1& {\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="120608142648.png"\; state="\{\{state\}\}">q</figure-callout>}}_2& \&CenterDot;\&CenterDot;\&CenterDot;& q_{n1}\\ r_1& {\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="120608142648.png"\; state="\{\{state\}\}">r</figure-callout>}}_2& \&CenterDot;\&CenterDot;\&CenterDot;& {\mathrm{<figure-callout\; id="1"\; label="r\; n"\; filenames="120608142648.png"\; state="\{\{state\}\}">r</figure-callout>}}_n\end{array}\right]=\left[\begin{array}{cccc}{\$}_1& {\$}_2& \&CenterDot;\&CenterDot;\&CenterDot;& {\$}_{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="120608142648.png"\; state="\{\{state\}\}">n</figure-callout>}1}\end{array}\right]$

Calculate the method in supplementary set space and find the solution following system of homogeneous linear equations exactly

S ^Tx=0 (2)

Non-0 of equation group is separated the basic spiral of having represented the supplementary set space, obtains spiral set M so respectively ₁And M ₂In the complementary spiral of spiral system, the set of these two groups complementary spirals is designated as Q respectively ₁And Q ₂, they promptly are respectively supplementary set space T ₁And T ₂Basic spiral;

(3.2) calculate T according to step (3.1) ₁And T ₂Union space T ₁∪ T ₂, i.e. union Q ₁∪ Q ₂In basic spiral;

(3.3) in 6 dimension spaces, calculate union space T ₁∪ T ₂The supplementary set space, be A ₁∩ A ₂, obvious (T ₁∪ T ₂) ∪ (A ₁∩ A ₂)=R ⁶, computational methods are to find the solution set Q according to (formula 2) ₁∪ Q ₂In the complementary spiral of each spiral, its set is designated as M ₁₂, then gather M ₁₂In 6-Rank (Q is arranged ₁∪ Q ₂) the bar spiral, they are A ₁∩ A ₂Basic spiral, the spiral space that these basic spirals are opened is exactly desired common factor space, the dimension Dim (A in common factor space ₁∩ A ₂)=Rank (M ₁₂);

(4) calculate the frees degree of encircling the parallel institution moving platforms: computing formula is more

DOF(MovingPlatform)=Rank(M ₁₂)+Rank(M ₃)-Rank(M ₁₂∪M ₃) (3)；

(5) arrange the driver of parallel institution: for any two side chains of parallel institution, the parallel institution drive arrangement scheme that does not contain redundant drive must meet the following conditions, the one, get rid of the kinematic pair of configuration driven device after, side chain kinematic screw set M ₁' the spiral rank of matrix Rank (M that constitutes ₁')<rank (M ₁), same Rank (M ₂')<rank (M ₂), the 2nd, get rid of the kinematic pair of configuration driven device after, two side chain kinematic screw union of sets collection M ₁' ∪ M ₂' the spiral rank of matrix Rank (M that constitutes ₁' ∪ M ₂')=Rank (M ₁∪ M ₂), the 3rd, get rid of the kinematic pair of configuration driven device after, the spiral rank of matrix Rank (M that two side chain kinematic screw union of sets collection constitute ₁' ∪ M ₂')=Rank (M ₁')+Rank (M ₂');

The isolated degree of freedom that (6) possibly exist in the identification side chain: for the mechanism that carries out drive arrangement according to step (5), get rid of the kinematic pair that each side chain contains driver, calculate the new spiral rank of matrix of each side chain then, be designated as Rank (M ² _i), remove a driver then, then side chain increases the kinematic screw of a former driver place kinematic pair, calculates newly-increased kinematic screw side chain spiral rank of matrix afterwards, is designated as Rank (M ³ _i), if Rank (M ³ _i)=Rank (M ² _i), explain that former driver place kinematic pair is an isolated degree of freedom, this driver should be got rid of really, if Rank (M ³ _i)>Rank (M ² _i); Explain that former driver place kinematic pair is not an isolated degree of freedom; This driver can not be removed, and repeats said process and all accomplishes check up to whole drivers, for no longer containing the mechanism of drive arrangement in isolated degree of freedom; Get rid of the kinematic pair that each side chain contains driver, calculate the new spiral rank of matrix Rank (M of each side chain then ⁴ _i), and calculate the free degree sum W of the last kinematic pair of side chain at this moment _i, the number ξ of isolated degree of freedom of each bar side chain then _i=W _i-Rank (M ⁴ _i);

(7) the whole free degree of calculating parallel institution: the whole free degree formula of mechanism of having considered redundant kinematic screw of side chain and isolated degree of freedom does

$\mathrm{DOF}\left(\mathrm{Mechanism}\right)=\mathrm{DOF}\left(\mathrm{MovingPlatform}\right)+\underset{i=1}{\overset{I}{\mathrm{\&Sigma;}}}{F}_{\mathrm{ri}}-\underset{i=1}{\overset{I}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i---\left(4\right)$

Wherein, the free degree of the redundant kinematic screw of side chain does

$F_{\mathrm{ri}}=\underset{j=1}{\overset{g_i}{\mathrm{\&Sigma;}}}{F}_{\mathrm{ij}}-\mathrm{Rank}\left(M_i\right)$

All degree of freedom of kinematic pair summations in

expression side chain.

Calculate the free degree of moving platform through the common factor space dimensionality in each side chain kinematic screw space of parallel institution.

Through the notion of complementary spiral, provided the detailed process of spiral space intersection operation.

For two side chain single loop mechanisms, need not to ask for the common factor space, and can directly obtain the moving platform free degree.

Dimension for two side chain spiral space unions equals a side chain spiral rank of matrix, perhaps equals the situation of two side chain spiral rank of matrix sums, can directly obtain to occur simultaneously the space and need not any computing.

The beneficial effect that the present invention has is:

1) the freedom calculation method that proposes of the present invention moving platform free degree and mechanism's integral body free degree of correct calculation all types parallel institution unlimitedly;

2) the freedom calculation method of the present invention's proposition need not to ask for the common factor space of two side chain spiral spaces, and can directly obtain the moving platform free degree for single loop mechanism;

3) the freedom calculation method that proposes of the present invention equals a side chain spiral rank of matrix for the dimension of two side chain spiral space unions, perhaps equals the situation of two side chain spiral rank of matrix sums, can directly obtain to occur simultaneously the space and need not any computing.

Description of drawings

Fig. 1 is based on the flow chart of the parallel institution freedom calculation of spiral space intersection operation.

Fig. 2 is a monocycle 6R parallel institution sketch map.

Fig. 3 is many ring 3-CPR parallel institution sketch mapes.

Among the figure: 1, cylindrical pair, 2, moving sets, 3, revolute pair, 4, cylindrical pair, 5, moving sets, 6, revolute pair, 7, cylindrical pair, 8, moving sets, 9, revolute pair.

The specific embodiment

To combine accompanying drawing and embodiment that the present invention is further described below.

(1) combine monocycle 6R mechanism that the present invention is described further.See shown in Figure 1ly based on the flow chart of the parallel institution freedom calculation method of spiral space intersection operation,, comprise following committed step for the single loop mechanism freedom calculation:

1) dimension of calculating side chain spiral space;

2) calculate two side chain single loop mechanism moving platform frees degree;

3) driver of layout parallel institution;

The isolated degree of freedom that 4) possibly exist in the identification side chain;

5) the whole free degree of calculating parallel institution.

Monocycle 6R mechanism with shown in Figure 2 is an example.Each kinematic pair of mechanism all is a revolute pair, is mechanism's moving platform with P point place rod member among Fig. 2, and the P point left side is first side chain, and P point the right is second side chain.

1) dimension of calculating side chain spiral space:

Because every side chain is formed by three revolute pairs, each revolute pair has one degree of freedom.According to screw theory, the spiral rank of matrix of first side chain is Rank (M ₁)=3, therefore the dimension of the first side chain spiral space is 3; The spiral rank of matrix Rank (M of second side chain ₂)=3, therefore the dimension of the second side chain spiral space also is 3.

2) calculate two side chain single loop mechanism moving platform frees degree:

Because the union spiral rank of matrix of first side chain and the second side chain spiral space is Rank (M ₁∪ M ₂)=3 have according to (formula 1), DOF (MovingPlatform)=Rank (M ₁)+Rank (M ₂)-Rank (M ₁∪ M ₂)=3+3-3=3, promptly mechanism's moving platform free degree is three.

3) driver of layout parallel institution:

Mechanism's moving platform has three degree of freedom, therefore needs to arrange three drivers.These three drivers can be arranged in any three joints of 6R parallel connection, all meet the demands.

The isolated degree of freedom that 4) possibly exist in the identification side chain:

In any case arrange three drivers, two side chains all do not contain redundant degree of freedom and isolated degree of freedom.

5) the whole free degree of calculating parallel institution:

Have according to (formula 4), $\mathrm{DOF}\left(\mathrm{Mechanism}\right)=\mathrm{DOF}\left(\mathrm{MovingPlatform}\right)+\underset{i=1}{\overset{I}{\mathrm{\&Sigma;}}}{F}_{\mathrm{Ri}}-\underset{i=1}{\overset{I}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i=$ $3+0-0=3,$ Be that the whole free degree of mechanism is 3.

(2) combine how ring 3-CPR mechanism is described further the present invention.Based on the flow chart of the parallel institution freedom calculation method of spiral space intersection operation, see shown in Figure 1ly, for the multi-loop mechanism freedom calculation, comprise following committed step:

1) dimension of calculating side chain spiral space;

2) the common factor space of two side chain spiral spaces of calculating;

3) calculate the frees degree of encircling the parallel institution moving platforms more;

4) driver of layout parallel institution;

The isolated degree of freedom that 5) possibly exist in the identification side chain;

6) the whole free degree of calculating parallel institution.

With shown in Figure 3 many ring 3-CPR mechanism is example.Three side chains are contained in mechanism among Fig. 3, and the left side is first side chain, and the centre is second side chain, and the right is the 3rd side chain.

1) dimension of calculating side chain spiral space:

Because every side chain is formed by 2,5,8, one revolute pairs of 1,4,7, one moving sets of a cylindrical pair 3,6,9, cylindrical pair has two frees degree, and moving sets has one degree of freedom, and revolute pair has one degree of freedom.According to screw theory, the spiral rank of matrix of first side chain is Rank (M ₁)=4, therefore the dimension of the first side chain spiral space is 4; The spiral rank of matrix Rank (M of second side chain ₂)=4, therefore the dimension of the second side chain spiral space also is 4; The spiral rank of matrix Rank (M of the 3rd side chain ₃)=4, therefore the dimension of the 3rd side chain spiral space also is 4.

2) the common factor space of two side chain spiral spaces of calculating:

Because Rank (M ₁∪ M ₂)=4, therefore Rank (M is contained in the common factor space of first side chain and second side chain ₁)+Rank (M ₂)-Rank (M ₁∪ M ₂)=4 a basic spiral is designated as A with its spiral space that becomes ₁₂, the spiral matrix is designated as M ₁₂Because N ₁=N ₂=N ₃=4, M is described ₂Middle spiral vector is to M ₁∪ M ₂The number of middle spiral vector is contribution, i.e. spiral space A not ₁∪ A ₂Be equal to spiral space A ₁Can certainly be interpreted as M ₁Middle spiral vector is to M ₁∪ M ₂The number of middle spiral vector is contribution, i.e. spiral space A not ₁∪ A ₂Be equal to spiral space A ₂Therefore, this moment A ₁₂=A ₁∩ A ₂=A ₁=A ₂, i.e. M ₁Perhaps M ₂Middle spiral vector is spiral space A ₁∩ A ₂Basic spiral, Rank (M ₁₂)=4.

3) calculate the frees degree of encircling the parallel institution moving platforms more:

Because Rank (M ₃)=4, Rank (M ₁₂∪ M ₃)=5, therefore according to (formula 3), the free degree DOF of moving platform (MovingPlatform)=Rank (M ₁₂)+Rank (M ₃)-Rank (M ₁₂∪ M ₃)=4+4-5=3.

4) driver of layout parallel institution:

The structure moving platform has three degree of freedom, therefore needs to arrange three drivers.Can only arrange a driver on the rotational freedom of the cylindrical pair of first side chain and second side chain; If arrange two drivers, then Rank (M simultaneously ₁' ∪ M ₂')=3<rank (M ₁∪ M ₂)=4 promptly produce and overdrive.Usually these 3 drive arrangement are on the one-movement-freedom-degree of the cylindrical pair of first side chain, second side chain, the 3rd side chain.

The isolated degree of freedom that 5) possibly exist in the identification side chain:

In any case arrange two drivers, two side chains all do not contain isolated degree of freedom.

6) the whole free degree of calculating parallel institution:

Have according to (formula 4), $\mathrm{DOF}\left(\mathrm{Mechanism}\right)=\mathrm{DOF}\left(\mathrm{MovingPlatform}\right)+\underset{i=1}{\overset{I}{\mathrm{\&Sigma;}}}{F}_{\mathrm{Ri}}-\underset{i=1}{\overset{I}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i=$ $3+0-0=3,$ Be that the whole free degree of mechanism is 3.

Claims (5)

1. parallel institution freedom analysis method based on the spiral space intersection operation is characterized in that the step of this method is following:

(1) dimension of calculating side chain spiral space: establish $ ⁱJ is the spiral vector of j kinematic pair in the i bar side chain, and then the set of the kinematic screw of each kinematic pair composition of side chain i is M _i={ $ ⁱ1, $ ⁱ2 ..., $ ⁱn _i, the rank of matrix that set constitutes is Rank (M _i)≤6, so the kinematic screw of side chain i constitutes Rank (M _i) the dimension spiral space, be designated as A _i, spiral space A _iDimension Dim (A _i)=Rank (M _i);

(2) calculate two side chain single loop mechanism moving platform frees degree: a side chain spiral space A ₁Dimension Dim (A ₁)=Rank (M ₁), another side chain spiral space A ₂Dimension Dim (A ₂)=Rank (M ₂), M ₁∪ M ₂={ $ ¹1, $ ¹2 ..., $ ¹n ₁, $ ²1, $ ²2 ..., $ ²n ₂, spiral space A then ₁And A ₂The dimension Dim (A in union space ₁∪ A ₂)=Rank (M ₁∪ M ₂), so monocycle parallel institution moving platform freedom calculation does

DOF(MovingPlatform)=Rank(M ₁)+Rank(M ₂)-Rank(M ₁∪M ₂) (1)；

(3) the common factor space of two side chain spiral spaces of calculating:

Kinematic screw set M according to two side chains<sub >1</sub>And M<sub >2</sub>And union M<sub >1</sub>∪ M<sub >2</sub>, calculate the spiral rank of matrix N that these set constitute<sub >1</sub>=Rank (M<sub >1</sub>), N<sub >2</sub>=Rank (M<sub >2</sub>), N<sub >3</sub>=Rank (M<sub >1</sub>∪ M<sub >2</sub>), suppose N<sub >1</sub>>=N<sub >2</sub>, obvious N<sub >3</sub>>=N<sub >1</sub>If, N<sub >3</sub>=N<sub >1</sub>, A then<sub >1</sub>∩ A<sub >2</sub>=A<sub >2</sub>, i.e. A<sub >2</sub>Be exactly spiral space A<sub >1</sub>∩ A<sub >2</sub>The common factor space, if N<sub >3</sub>=N<sub >1</sub>+ N<sub >2</sub>, M is described<sub >2</sub>In all spiral vectors to M<sub >1</sub>∪ M<sub >2</sub>The number of middle spiral vector all has contribution, i.e. spiral space A<sub >1</sub>With spiral space A<sub >2</sub>Do not comprise mutually, at this moment<img file="FDA00001742003100011.GIF" he="44" id="ifm0001" img-content="drawing" img-format="GIF" inline="yes" orientation="portrait" wi="293" />Be that mechanism can not move, if N<sub >1</sub><n<sub >3</sub><n<sub >1</sub>+ N<sub >2</sub>, M is described<sub >2</sub>Middle part screw amount is to M<sub >1</sub>∪ M<sub >2</sub>The number of middle spiral vector has contribution, A<sub >1</sub>∩ A<sub >2</sub>Basic spiral number in the spiral space is N<sub >1</sub>+ N<sub >2</sub>-N<sub >3</sub>, need calculate A this moment as follows<sub >1</sub>∩ A<sub >2</sub>Basic spiral:

(3.1) in 6 dimension spaces, calculate spiral space A respectively ₁And A ₂The supplementary set space, be designated as T ₁And T ₂, obvious T ₁∪ A ₁=T ₂∪ A ₂=R ⁶, use as if spiral is vectorial (l, m, n, p, q, r) ^TExpression, then the spiral matrix that constituted of spiral set does

$S=\left[\begin{array}{cccc}{l}_1& l_2& \&CenterDot;\&CenterDot;\&CenterDot;& l_{n1}\\ m_1& m_2& \&CenterDot;\&CenterDot;\&CenterDot;& m_{n1}\\ n_1& n_2& \&CenterDot;\&CenterDot;\&CenterDot;& n_{n1}\\ p_1& p_2& \&CenterDot;\&CenterDot;\&CenterDot;& p_{n1}\\ q_1& q_2& \&CenterDot;\&CenterDot;\&CenterDot;& q_{n1}\\ r_1& r_2& \&CenterDot;\&CenterDot;\&CenterDot;& r_{n1}\end{array}\right]=\left[\begin{array}{cccc}{\$}_1& {\$}_2& \&CenterDot;\&CenterDot;\&CenterDot;& {\$}_{n1}\end{array}\right]$

Calculate the method in supplementary set space and find the solution following system of homogeneous linear equations exactly

S ^Tx=0 (2)

Non-0 of equation group is separated the basic spiral of having represented the supplementary set space, obtains spiral set M so respectively ₁And M ₂In the complementary spiral of spiral system, the set of these two groups complementary spirals is designated as Q respectively ₁And Q ₂, they promptly are respectively supplementary set space T ₁And T ₂Basic spiral;

(3.2) calculate T according to step (3.1) ₁And T ₂Union space T ₁∪ T ₂, i.e. union Q ₁∪ Q ₂In basic spiral;

(3.3) in 6 dimension spaces, calculate union space T ₁∪ T ₂The supplementary set space, be A ₁∩ A ₂, obvious (T ₁∪ T ₂) ∪ (A ₁∩ A ₂)=R ⁶, computational methods are to find the solution set Q according to (formula 2) ₁∪ Q ₂In the complementary spiral of each spiral, its set is designated as M ₁₂, then gather M ₁₂In 6-Rank (Q is arranged ₁∪ Q ₂) the bar spiral, they are A ₁∩ A ₂Basic spiral, the spiral space that these basic spirals are opened is exactly desired common factor space, the dimension Dim (A in common factor space ₁∩ A ₂)=Rank (M ₁₂);

(4) calculate the frees degree of encircling the parallel institution moving platforms: computing formula is more

DOF(MovingPlatform)=Rank(M ₁₂)+Rank(M ₃)-Rank(M ₁₂∪M ₃) (3)；

(5) arrange the driver of parallel institution: for any two side chains of parallel institution, the parallel institution drive arrangement scheme that does not contain redundant drive must meet the following conditions, the one, get rid of the kinematic pair of configuration driven device after, side chain kinematic screw set M ₁' the spiral rank of matrix Rank (M that constitutes ₁')<rank (M ₁), same Rank (M ₂')<rank (M ₂), the 2nd, get rid of the kinematic pair of configuration driven device after, two side chain kinematic screw union of sets collection M ₁' ∪ M ₂' the spiral rank of matrix Rank (M that constitutes ₁' ∪ M ₂')=Rank (M ₁∪ M ₂), the 3rd, get rid of the kinematic pair of configuration driven device after, the spiral rank of matrix Rank (M that two side chain kinematic screw union of sets collection constitute ₁' ∪ M ₂')=Rank (M ₁')+Rank (M ₂');

The isolated degree of freedom that (6) possibly exist in the identification side chain: for the mechanism that carries out drive arrangement according to step (5), get rid of the kinematic pair that each side chain contains driver, calculate the new spiral rank of matrix of each side chain then, be designated as Rank (M ² _i), remove a driver then, then side chain increases the kinematic screw of a former driver place kinematic pair, calculates newly-increased kinematic screw side chain spiral rank of matrix afterwards, is designated as Rank (M ³ _i), if Rank (M ³ _i)=Rank (M ² _i), explain that former driver place kinematic pair is an isolated degree of freedom, this driver should be got rid of really, if Rank (M ³ _i)>Rank (M ² _i); Explain that former driver place kinematic pair is not an isolated degree of freedom; This driver can not be removed, and repeats said process and all accomplishes check up to whole drivers, for no longer containing the mechanism of drive arrangement in isolated degree of freedom; Get rid of the kinematic pair that each side chain contains driver, calculate the new spiral rank of matrix Rank (M of each side chain then ⁴ _i), and calculate the free degree sum W of the last kinematic pair of side chain at this moment _i, the number ξ of isolated degree of freedom of each bar side chain then _i=W _i-Rank (M ⁴ _i);

(7) the whole free degree of calculating parallel institution: the whole free degree formula of mechanism of having considered redundant kinematic screw of side chain and isolated degree of freedom does

$\mathrm{DOF}\left(\mathrm{Mechanism}\right)=\mathrm{DOF}\left(\mathrm{MovingPlatform}\right)+\underset{i=1}{\overset{I}{\mathrm{\&Sigma;}}}{F}_{\mathrm{ri}}-\underset{i=1}{\overset{I}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i---\left(4\right)$

Wherein, the free degree of the redundant kinematic screw of side chain does

$F_{\mathrm{ri}}=\underset{j=1}{\overset{g_i}{\mathrm{\&Sigma;}}}{F}_{\mathrm{ij}}-\mathrm{Rank}\left(M_i\right)$

All degree of freedom of kinematic pair summations in

expression side chain.

2. a kind of parallel institution freedom analysis method based on the spiral space intersection operation according to claim 1 is characterized in that: the free degree of calculating moving platform through the common factor space dimensionality in each side chain kinematic screw space of parallel institution.

3. a kind of parallel institution freedom analysis method based on the spiral space intersection operation according to claim 1 is characterized in that: through the notion of complementary spiral, provided the detailed process of spiral space intersection operation.

4. a kind of parallel institution freedom analysis method based on the spiral space intersection operation according to claim 1 is characterized in that: for two side chain single loop mechanisms, need not to ask for the common factor space, and can directly obtain the moving platform free degree.

5. a kind of parallel institution freedom analysis method according to claim 1 based on the spiral space intersection operation; It is characterized in that: the dimension for two side chain spiral space unions equals a side chain spiral rank of matrix; Perhaps equal the situation of two side chain spiral rank of matrix sums, can directly obtain to occur simultaneously the space and need not any computing.

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