CN106113042B - The geometry solving method of parallel institution instantaneous axis - Google Patents

The geometry solving method of parallel institution instantaneous axis Download PDF

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CN106113042B
CN106113042B CN201610590067.4A CN201610590067A CN106113042B CN 106113042 B CN106113042 B CN 106113042B CN 201610590067 A CN201610590067 A CN 201610590067A CN 106113042 B CN106113042 B CN 106113042B
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space
parallel institution
constraint
degree
freedom
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CN106113042A (en
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庹华
段志祥
韩建欢
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Rokae (beijing) Technology Co Ltd
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Rokae (beijing) Technology Co Ltd
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1615—Programme controls characterised by special kind of manipulator, e.g. planar, scara, gantry, cantilever, space, closed chain, passive/active joints and tendon driven manipulators

Abstract

The present invention proposes a kind of geometry solving method of parallel institution instantaneous axis, includes the following steps：The structure diagram of parallel institution is drawn out, and draws out the kinematic pair of each branch；Calculate the degree of freedom space of each branch；According to Generalized Dual rule, the constraint space of each branch is solved；According to the constraint space of each branch, parallel institution moving platform constraint space is solved；Further according to Generalized Dual rule, the instantaneous axis of solution parallel institution moving platform.The present invention proposes a kind of geometry solving method of parallel institution instantaneous axis, and complicated algebraic operation is converted for simple geometric operation, reduces computational complexity.Make result of calculation more directly perceived.

Description

The geometry solving method of parallel institution instantaneous axis
Technical field
The present invention relates to path adaptation technical field, more particularly to a kind of geometry solving method of parallel institution instantaneous axis.
Background technology
Instantaneous axis is a very important concept in parallel robot.It can be in the hope of the rotation axis of its moving platform by instantaneous axis Line and speed.At present, the method that the solution of instantaneous axis uses algebraically mostly, main method have：Pass through Jacobian matrix and input speed It spends to solve the instantaneous axis of moving platform and solve by attitude matrix.
It is solved by the method for Jacobi：
Wherein, J is that the spinor of parallel robot moving platform is expressed, and ω is the Jacobian matrix of robot,For input angle speed Degree.Instantaneous axis is solved by the method for attitude matrix：
Wherein, A represents the matrix of one 4 × 4, is the attitude matrix of moving platform in parallel, andFor the derivative of the matrix, A-1 Meet AA-1=Ι.
The major defect and deficiency that can be seen that the method for solving of existing instantaneous axis from above two method for solving are： Calculation amount is bigger, and solving result is not directly perceived enough.
Such as the patent of invention of Chinese patent CN 102107431A, which provides a kind of parallel robot, however, the hair The major defect and deficiency of the method for solving of bright parallel robot instantaneous axis are：Calculation amount is bigger, and solving result is not straight enough It sees.
The content of the invention
The purpose of the present invention is intended at least solve one of described technological deficiency.
For this purpose, it is an object of the invention to propose a kind of geometry solving side of solving result intuitively parallel institution instantaneous axis Method.
To achieve these goals, the present invention provides a kind of geometry solving method of parallel institution instantaneous axis, including walking as follows Suddenly：
Step S1 according to the structure of parallel institution, draws out the structure diagram of parallel institution, and draws out each branch Kinematic pair；
Step S2 according to the kinematic pair of drawn out each branch, calculates the degree of freedom space of each branch；
Step S3 according to Generalized Dual rule, solves the constraint space of each branch；
Step S4 according to the constraint space of each branch, solves parallel institution moving platform constraint space；
Step S5, further according to Generalized Dual rule, the instantaneous axis of solution parallel institution moving platform.
Further, in step sl, when drawing out the kinematic pair of each branch, using different colours, different directions, have The straight line of no arrow distinguishes.
Further, in step s 2, the degree of freedom space is the set for including multiple kinematic pair spinor lines；
One vector $ in the degree of freedom spacefIt is expressed as：
$f=$1∩$2∩...∩$i...∩$n；
Wherein, n is all branch kinematic pair number summations, and $ represents the spinor of kinematic pair.
Further, in step s 2, the corresponding spinor of revolute pair is expressed as：
Wherein, s is the unit vector of spinor axis direction, can be represented with the cosine in three directions, and r is the spinor axis On any point；So, corresponding geometric expression is a line, which is the axis by revolute pair.
Further, in step s 2, the corresponding spinor of prismatic pair is expressed as：
Wherein, s represents the direction of prismatic pair, and geometric expression carries the line of arrow for both ends；
Further, the Generalized Dual rule meets the following conditions：
A, every rotational freedom line in degree of freedom space all with force constraint line phase all in its dual constraint space It hands over or parallel；Vice versa；
B, every moving direction line in degree of freedom space is all vertical with force constraint line all in its dual constraint space； Conversely, the even amount normal of each constraint in constraint space hangs down with all rotational freedom lines in its antithesis degree of freedom space Directly；
C, occasionally amount direction line can be arbitrary with the constraint in its dual constraint space for the moving direction line in degree of freedom space Configuration；
The general motion axis of screw in degree of freedom space meets with the general wrench of a force system axis in its dual constraint space：
pF+pC=dFCtanαFC, F=1,2 ..., n；C=1,2 ..., 6-n；
Wherein pFFor the pitch of kinematic screw in degree of freedom space, pCTo constrain the pitch of the wrench of a force system, d in constraint spaceFC For the common vertical line distance of two spirals, αFCFor the angle of two spirals.
Further, meet antithesis according to the vector in the vector sum degree of freedom space in Generalized Dual rule constraint space to close System.I.e.：
Wherein, $fOne vector in expression degree of freedom space, and $cIt is the vector in constraint space.
Further, in step s 5, according to Generalized Dual rule, acquire and the vector $ in constraint spacecThe arrow of antithesis Measure $1,$2,$3...$i, then using parallelogram law, the vector obtained to above-mentioned solution synthesizes, then after synthesizing Vector is the instantaneous axis of parallel institution moving platform.
The present invention proposes a kind of geometry solving method of parallel institution instantaneous axis, and complicated algebraic operation is converted in order to simple Geometric operation, reduce computational complexity.Make result of calculation more directly perceived.
The additional aspect of the present invention and advantage will be set forth in part in the description, and will partly become from the following description It obtains substantially or is recognized by the practice of the present invention.
Description of the drawings
The above-mentioned and/or additional aspect and advantage of the present invention will become in the description from combination accompanying drawings below to embodiment Substantially and it is readily appreciated that, wherein：
Fig. 1 a- Fig. 1 g are the geometric expression schematic diagram of common kinematic pair；
Fig. 2 is the overall flow figure of the present invention；
Fig. 3 is parallelogram law schematic diagram；
Fig. 4 is instantaneous axis geometry solving flow chart of the present invention.
Specific embodiment
The embodiment of the present invention is described below in detail, the example of the embodiment is shown in the drawings, wherein from beginning to end Same or similar label represents same or similar element or has the function of same or like element.Below with reference to attached The embodiment of figure description is exemplary, it is intended to for explaining the present invention, and is not considered as limiting the invention.
Method of geometry is the related operation based on spinor, is a kind of geometric expression of algebraic operation in spinor.
Some basic concepts are introduced first, parallel institution generally has several branches, can all have kinematic pair on every branch, The geometric expression of each kinematic pair is introduced first：
For specific revolute pair (R) in mechanism, only 1 degree of freedom, with the red to coincide with pivot center Straight line characterizes, such as shown in (Fig. 1 a), straight line of the both ends without arrow not only there is direction attribute as pivot center and And also there is position attribution；In mechanism common prismatic pair (P) also only have 1 degree of freedom, with one it is parallel with moving direction and Both ends straight line with the arrow characterizes, such as (shown in (Fig. 1 b), since moving movement only has direction attribute, so the freedom of movement Spending line equally only has direction attribute.
The common pure rolling of one kind is secondary in mechanism, such as meshed gears is secondary, friction free cam pair, and this kinematic pair also only has 1 degree of freedom, transient motion are considered as two Objects around A contact lines and do relatively pure rolling, can use one and contact line weight Red straight line of the both ends of conjunction without arrow characterizes, such as (shown in (Fig. 1 c), which represents that symbol is the same not only with rotational motion With direction attribute but also with position attribution.
Common Hooke's hinge (U) has 2 rotational freedoms in mechanism, such as (shown in (Fig. 1 d), respectively with two respectively with Pivot center overlaps and intersecting straight line of the both ends without arrow characterizes, and the intersection point of two straight lines is overlapped with Hooke's hinge center；Mechanism In common cylindrical pair (C) have mobile and rotate 2 degree of freedom, such as shown in (Fig. 1 e), rotated respectively with one and cylindrical pair The both ends that axis overlaps characterize its rotational motion without arrow straight line, then with a both ends parallel with cylindrical pair pivot center with arrow Head straight line characterizes its moving movement.
Common spherical pair (S) has 3 rotational freedoms in mechanism, such as shown in (Fig. 1 f), with three non-coplanar but remittances The both ends for meeting at space common point represent that the space common point is overlapped with spherical pair central point without arrow straight line；It is common in mechanism Planar contact pair (E) there are 1 rotational freedom and 2 one-movement-freedom-degrees, such as shown in (Fig. 1 g), with two with plane parallel but phase The both ends straight line with the arrow being not parallel to each other and one represent with the perpendicular both ends of plane without arrow straight line.
The geometric expression is corresponded with their spinor algebra expression.
The present invention provides a kind of geometry solving method of parallel institution instantaneous axis, and refer to the attached drawing 2-3 includes the following steps：
Step S1 according to the structure of parallel institution, draws out the structure diagram of parallel institution, and draws out each branch Kinematic pair；
When drawing out the kinematic pair of each branch, different colours, different directions, the straight line progress area for whetheing there is arrow are utilized Point.
Movement and the collection of illustrative plates symbolic indication of power
For example, rotating symbol represents that the shaft of the straight line and revolute pair coincides with straight line.Prismatic pair is with the arrow Line represents that direction is consistent with mobile direction.
Step S2 according to the kinematic pair of drawn out each branch, calculates the degree of freedom space of each branch；
The line chart constitutes the degree of freedom space of each branch, similar to the formation of vector space in linear algebra, freely Degree space refers to the space being turned by object of which movement spinor, is the set for including multiple movement spinor lines, it characterizes object Permitted spatial movement seeks common ground to each revolute pair and can obtain degree of freedom space.
One vector $ in degree of freedom spacefIt is expressed as：
$f=$1∩$2∩...∩$i...∩$n, (1)；
Wherein, n is all branch kinematic pair number summations, and $ represents the spinor of kinematic pair.
The corresponding spinor of revolute pair is expressed as：
Wherein, s is the unit vector of spinor axis direction, can be represented with the cosine in three directions, and r is the spinor axis On any point；So, corresponding geometric expression is a line, which is the axis by revolute pair.
It may determine that whether painted revolute pair spinor line is correct using above-mentioned formula (2).
The corresponding spinor of prismatic pair is expressed as：
Wherein, s represents the direction of prismatic pair, and geometric expression carries the line of arrow for both ends；
It may determine that whether painted revolute pair spinor line is correct using above-mentioned formula (3).
Other kinematic pairs can be regarded as the linear combination of both basic exercises pair.
Step S3 according to Generalized Dual rule, solves the constraint space of each branch；
Generalized Dual rule meets the following conditions：
A, every rotational freedom line in degree of freedom space all with force constraint line phase all in its dual constraint space It hands over or parallel；Vice versa；
B, every moving direction line in degree of freedom space is all vertical with force constraint line all in its dual constraint space； Conversely, the even amount normal of each constraint in constraint space hangs down with all rotational freedom lines in its antithesis degree of freedom space Directly；
C, occasionally amount direction line can be arbitrary with the constraint in its dual constraint space for the moving direction line in degree of freedom space Configuration；
The general motion axis of screw in degree of freedom space meets with the general wrench of a force system axis in its dual constraint space：
pF+pC=dFC tanαFC, F=1,2 ..., n；C=1,2 ..., 6-n； (4)；
Wherein pFFor the pitch of kinematic screw in degree of freedom space, pCTo constrain the pitch of the wrench of a force system, d in constraint spaceFC For the common vertical line distance of two spirals, αFCFor the angle of two spirals.
Step S4 according to the constraint space of each branch, solves parallel institution moving platform constraint space；
Duality relation is met according to the vector in the vector sum degree of freedom space in Generalized Dual rule constraint space.Constrain The product of the vector in the vector sum degree of freedom space in space is 0：
Formula is：
Wherein, $fOne vector in expression degree of freedom space, and $cIt is the vector in constraint space.
Step S5, further according to Generalized Dual rule, the instantaneous axis of solution parallel institution moving platform.
According to Generalized Dual rule, acquire and the vector $ in constraint spacecThe vector $ of antithesis1,$2,$3...$i, Ran Houli With parallelogram law, as shown in figure 4, the vector obtained to above-mentioned solution synthesizes, then the vector after synthesizing is parallel connection The instantaneous axis of mechanism moving platform.
The present invention proposes a kind of geometry solving method of parallel institution instantaneous axis, and complicated algebraic operation is converted in order to simple Geometric operation, reduce computational complexity.Make result of calculation more directly perceived.
Although the embodiment of the present invention has been shown and described above, it is to be understood that above-described embodiment is example Property, it is impossible to limitation of the present invention is interpreted as, those of ordinary skill in the art are not departing from the principle of the present invention and objective In the case of above-described embodiment can be changed within the scope of the invention, change, replace and modification.The scope of the present invention It is extremely equally limited by appended claims.

Claims (8)

  1. A kind of 1. geometry solving method of parallel institution instantaneous axis, which is characterized in that include the following steps：
    Step S1 according to the structure of parallel institution, draws out the structure diagram of parallel institution, and draws out the movement of each branch It is secondary；
    Step S2 according to the kinematic pair of drawn out each branch, calculates the degree of freedom space of each branch；
    Step S3 according to Generalized Dual rule, solves the constraint space of each branch；
    Step S4 according to the constraint space of each branch, solves parallel institution moving platform constraint space；
    Step S5, further according to Generalized Dual rule, the instantaneous axis of solution parallel institution moving platform.
  2. 2. a kind of geometry solving method of parallel institution instantaneous axis as described in claim 1, it is characterised in that：In step sl, When drawing out the kinematic pair of each branch, distinguished using different colours, different directions, the straight line that whether there is arrow.
  3. 3. a kind of geometry solving method of parallel institution instantaneous axis as described in claim 1, it is characterised in that：In step s 2, The degree of freedom space is the set for including multiple kinematic pair spinor lines；
    One vector $ in the degree of freedom spacefIt is expressed as：
    $f=$1∩$2∩...∩$i...∩$n；
    Wherein, n is all branch kinematic pair number summations, and $ represents the spinor of kinematic pair.
  4. 4. a kind of geometry solving method of parallel institution instantaneous axis as described in claim 1, it is characterised in that：In step s 2, The corresponding spinor of revolute pair is expressed as：
    <mrow> <mi>$</mi> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mi>s</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>r</mi> <mo>&amp;times;</mo> <mi>s</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    Wherein, s is the unit vector of spinor axis direction, can be represented with the cosine in three directions, and r is on the spinor axis Any point；So, corresponding geometric expression is a line, which is the axis by revolute pair.
  5. 5. a kind of geometry solving method of parallel institution instantaneous axis as described in claim 1, it is characterised in that：In step s 2, The corresponding spinor of prismatic pair is expressed as：
    <mrow> <mi>$</mi> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    Wherein, s represents the direction of prismatic pair, and geometric expression carries the line of arrow for both ends.
  6. 6. a kind of geometry solving method of parallel institution instantaneous axis as described in claim 1, it is characterised in that：The Generalized Dual Rule meets the following conditions：
    A, every rotational freedom line in degree of freedom space all intersect with force constraint line all in its dual constraint space or It is parallel；Vice versa；
    B, every moving direction line in degree of freedom space is all vertical with force constraint line all in its dual constraint space；Instead It, the even amount normal of each constraint in constraint space is vertical with all rotational freedom lines in its antithesis degree of freedom space；
    C, occasionally amount direction line can be with arbitrary disposition with the constraint in its dual constraint space for the moving direction line in degree of freedom space；
    The general motion axis of screw in degree of freedom space meets with the general wrench of a force system axis in its dual constraint space：
    pF+pC=dFCtanαFC, F=1,2 ..., n；C=1,2 ..., 6-n；
    Wherein pFFor the pitch of kinematic screw in degree of freedom space, pCTo constrain the pitch of the wrench of a force system, d in constraint spaceFCFor two spiral shells The common vertical line distance of rotation, αFCFor the angle of two spirals.
  7. 7. a kind of geometry solving method of parallel institution instantaneous axis as described in claim 1, it is characterised in that：According to Generalized Dual The vector in the vector sum degree of freedom space in rule constraint space meets duality relation, i.e.,：
    Wherein, $fOne vector in expression degree of freedom space, and $cIt is the vector in constraint space.
  8. 8. a kind of geometry solving method of parallel institution instantaneous axis as described in claim 1, it is characterised in that：In step s 5, According to Generalized Dual rule, acquire and the vector $ in constraint spacecThe vector $ of antithesis1,$2,$3...$i, then utilize parallel four Side shape rule, the vector obtained to above-mentioned solution synthesize, then the vector after synthesizing is the instantaneous axis of parallel institution moving platform.
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CN112936289B (en) * 2021-03-24 2022-11-04 太原科技大学 Method for judging virtual constraints and number of virtual constraints in mechanism based on continuity of overconstrained

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CN103472724A (en) * 2013-09-16 2013-12-25 江苏大学 Real-time control dynamics modeling method for multi-freedom-degree parallel mechanism
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