CN109101033A - A kind of six free platform stance normal solution methods based on crank link mechanism - Google Patents
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Abstract
The present invention provides a kind of posture normal solution method of six degree of freedom platform, suitable for the six degree of freedom platform based on crank link mechanism.Firstly, establish platform base coordinate system, determine that the drive shaft of six driving motors or its transmission mechanism connect the center point coordinate of bearing with crank；It is then determined the bearing centre coordinate of crank and connecting rod junction；Then, according to the length of connecting rod, the geometrical relationship of each supporting point of upper mounting plate, one group of linear equation of platform and supporting point is obtained；Finally, solve system of equation obtains posture and the position of upper mounting plate.Method provided by the present invention is able to solve the six degree of freedom platform stance Positive Solutions based on crank link mechanism.Using on pedestal and upper mounting plate bearing tie point, crank, connecting rod geometrical relationship column write equation group, solve system of equation, the triaxial attitude angle and relative position for obtaining upper mounting plate have obtained the posture normal solution method of the six degree of freedom platform of brace and connecting rod framework.
Description
Technical field
The present invention relates to six degree of freedom platform courses technical field, in particular to a kind of six based on crank link mechanism are certainly
By platform stance normal solution method.
Background technique
Six degree of freedom platform refers to the effect by six supporting mechanisms in parallel, is able to carry out six free movement of space
Platform.Robot, various carriers motion state emulation, game in terms of have broad application prospects.
The free platform of current six mostly uses six supporting mechanisms with Telescopic as movement executing mechanism, typical
Structure such as stewart platform (D.Stewart. " A Platform with six degrees of freedom. "
Proeeedings of the IMeehE,1965,180(15):371-385).The six degree of freedom platform of this structure mostly uses
Hydraulic cylinder or electric cylinder driving, but structure is complicated, cost is high, volume is big, maintenance is inconvenient for both drive systems.California, USA
Two student Tylet of the School of Mechanical Engineering of San Jose state university (San Jose State University, SJSU)
Kroymann and Robert Dee has developed in term project Full Motion Dynamic (FMD) in 2013
A kind of six degree of freedom platform of the mechanism based on crank connecting link.This platform structure is simple, dynamic characteristic is good, causes extensive
Concern.The mechanisms such as domestic BJ University of Aeronautics & Astronautics, the big Hua Jiekang Science and Technology Ltd. in Beijing and personal also structure application accordingly
The patent of relevant six degree of freedom dynamic platform.(Xie Feng, Hu Lei, in knowing bright equal " a kind of six-freedom parallel vivid platform ",
Patent of invention, application number: 201610373911.8；Hu Lei, Liu Hongsheng, it is hard equal " a kind of based on six degree of freedom body-sensing platform
Skiing simulating machine ", utility model patent, application number: 201720903740.5；Tang Shanshan, " for analog simulation two, three,
Four, 6-dof motion platform ", utility model patent, application number: 201620888673.X).
The kinematics analysis of six degree of freedom platform is the key that promote six degree of freedom platform property skill under various application scenarios
Art, it is platform stance error analysis, power credit that wherein posture normal solution, which is a part indispensable in movement control technology,
The important technology that analysis, fault recovery, mechanism size optimization etc. require.The posture normal solution of six degree of freedom platform is
Refer to that the current state according to six supporting mechanisms in parallel solves the current posture and position of platform, for stewart platform
Speech is posture and the position that platform is solved according to the length of six telescopic rods；And for the six degree of freedom based on crank link mechanism
Platform is then the posture that the angle position information current according to crank seeks platform.Six degree of freedom platform stance is counter, and to solve problem simpler
It is single, and Positive Solutions are then extremely complex, and with ambiguity, (appoint auspicious, " 6DOF gesture stability Platform Key Technologies are ground
Study carefully ", China Engineering Physics Research Institute's doctoral thesis, in August, 2012；The Wu Peidong, " kinematics and reserve motion power of stewart platform
Basic research ", East China University of Science and Technology doctoral thesis, in September, 2008).For the six degree of freedom of common stewart structure
The common method of platform, posture normal solution has: iterative method or optimization, analytic method, Homotopy Method, Mathematics Mechanical method and neural network
It is proposed Deng, some scholars and particle swarm optimization algorithm is shunk based on ratio regenerative space (Li Lei, " degree-of-freedom platform position is just
Solution and control method research ", Harbin Engineering University's doctoral thesis, in July, 2008).Above method is all based on telescopic rod knot
The six degree of freedom platform of structure, that is, stewart structure platform posture normal solution method.
At present the documents and materials do not published still to the posture of the six degree of freedom platform based on crank link mechanism just
Solution method.
Summary of the invention
The technical problem to be solved by the present invention is the six degree of freedom platform stance Positive Solutions based on crank link mechanism,
Angle position information i.e. current according to crank seeks the posture of platform.
The technical proposal adopted by the invention to solve the above technical problems is that: firstly, platform base coordinate system is established, according to
Mechanical structure determines the position coordinates of six driving motors or its drive mechanism and crank junction；Then, current according to crank
Angle Position and crank length determine the spring bearing centre coordinate of crank Yu connecting rod junction；Then, according to the length of connecting rod
The geometrical relationship of degree, each supporting point of upper mounting plate, obtains one group of linear equation of platform and supporting point；Finally, solve system of equation,
Obtain posture and the position of upper mounting plate.
The mechanical structure of six degree of freedom platform as indicated with 1, fix relative to ground by pedestal, is equipped with six motors thereon；
Motor directly drives or drives six crank-motions by speed reducer；The cranked rotation axis of institute in the same plane, and is handed over
In a point o, crank drives connecting rod by spring bearing；Connecting rod is connected by six supporting points of spring bearing and upper mounting plate, band
Dynamic upper mounting plate movement.
The specific implementation step of this method are as follows:
(1) platform base coordinate system E is established0, six driving motors or the drive of its transmission mechanism are determined according to mechanical structure
Moving axis connect the center point coordinate of bearing with crank.Specific practice is as follows:
Establish platform base coordinate system E0It is as shown in Fig. 2: using zenith direction as z-axis positive direction；With six axis driving motors
Or the plane where speed reducer rotation axis is the reference planes (also referred to as xoy plane) of z=0；Six crank connecting links and motor turn
The supporting point of shaft connection place constitutes a hexagon, and taking six crossing point of axes is coordinate origin o；Be immediately ahead of platform y-axis just
Direction, right-hand direction are positive direction of the x-axis.Rotation angle around reference axis ox axis is defined as pitching angle theta, around the rotation angle of oy axis
It is defined as roll angle γ, the rotation angle around oz axis is defined as yaw angle
Defining crank and connecting the central point of bearing with the drive shaft of motor or its transmission mechanism is supporting point Ai(i=1 ...,
6), six supporting points constitute a hexagon A1A2A3A4A5A6.The coordinate A of six supporting points is determined according to mechanical structurei(xAi,
yAi,zAi) (i=1 ..., 6).
(2) according to crank current angle position, the spring bearing centre coordinate of crank Yu connecting rod junction is determined.Specific practice
It is as follows:
As shown in figure 3, definition and supporting point AiThe crank LA of connectioniIn coordinate system E0When reference planes (xoy plane) are interior
For initial position, LA is rememberediInitial position phasorRemember LAiCurrent time position With initial positionAngle
αi, (i=1~6 are positive counterclockwise)；The bearing centre point D of crank and connecting rod junctioni；It is coupled connecting rod and upper mounting plate junction
Spring bearing central point be Bi(i=1 ..., 6)；Crank LAiRotary shaft be vectorCrank initial position
And rotating vector piIt is distributed in reference planes as shown in figure 4, as seen from the figure: pi=[xAi,yAi,zAi]。
It is gained knowledge by movement it is found that any vector surrounds piRotate angle [alpha]iSpin matrix expression formula are as follows:
Z=cos αiI+(1-cosαi)pipi+sinαiPi (1)
P in above formulaiFor vector piCorresponding coordinate battle array, pipiFor vector piThe expression formula of corresponding dyad, the two is respectively as follows:
Crank LAiPostrotational vectorExpression formula are as follows:
And then available DiCoordinate (the x of pointdi,ydi,zdi):
(3) according to the length of connecting rod, the geometrical relationship of each supporting point of upper mounting plate, one group of line of platform and supporting point is obtained
Property equation.Specific practice is as follows:
Define each spring bearing centre of sphere central point of upper mounting plate are as follows: Bi(i=1 ..., 6), BiIn coordinate system E0In coordinate be
(xbi,ybi,zbi)。BiSix point distributions are as shown in figure 5, this six points constitute hexagon B1B2B3B4B5B6, hexagon long side length
|B1B2|=| B3B4|=| B5B6|=lb, bond length | B2B3|=| B4B5|=| B6B1|=hb, the extended line of three long sides meets at
Three point Cb1、Cb2、Cb3, these three points are in coordinate system E0In coordinate be (xcj,ycj,zcj) (j=1,2,3)., it is clear thatFor
One equilateral triangle.
By Fig. 5 it is known that triangleSide CbjCbkThe length of (j=1,2,3k=1,2,3 and j ≠ k) are as follows:
|CbjCbk|=2hb+lb (6)
Bi、DiWith adjacent CbjMay be constructed a triangle, if Bi、DiBetween connecting rod be LBi, vector expression
Formula isLength is LB, then according to Bi、Di、CbjCorrelation it is available:
Because | CbjBi|=hbIf k=hb/|CbjCbk|=hb/2hb+lb, (7) formula can be deformed to obtain:
So having:
It is available about C according to formula 9b1、Cb2、Cb3Three coordinate (xcj,ycj,zcj) (j=1,2,3) six sides
Journey:
Further according to | CbjCbk|=LC(j=1,2,3, k=1,2,3 and j ≠ k), can arrange and write triangleSide length
Formula is obtained about (xcj,ycj,zcj) (j=1,2,3) the other three equation:
(xc1-xc2)2+(yc1-yc2)2+(zc1-zc2)2-Lc 2=0 (16)
(10)~(18) formula in this way is constituted about xc1、yc1、zc1、xc2、yc2、zc2、xc3、yc3、zc3This nine unknown numbers
Equation group.
(4) solve system of equation obtains posture and the position of upper mounting plate.Specific practice is as follows:
The equation group that this nine equations of (10)~(18) are constituted is solved using the method for solving Nonlinear System of Equations, can be obtained
To xc1、yc1、zc1、xc2、yc2、zc2、xc3、yc3、zc3This nine unknown numbers.
The upper mounting plate coordinate system E that definition is connected firmly with upper mounting plateb, the definition of the coordinate system is as shown in Figure 7: EbReference planes
(zb=0 plane) take six spring bearing center point B of upper mounting platei(i=1 ..., 6) place plane；Origin obIt is several to be taken as upper mounting plate
What center；When six cranks are in horizontal position (i.e. α shown in FIG. 1i=0, i=1,2 ..., 6) upper mounting plate is in initial shape when
State, o under original statebxbAxis and platform base coordinate system E0Middle ox axis is parallel；obybAxis and platform base coordinate system E0Middle oy axis
In parallel；At this point, EbCoordinate origin distance E0The height of coordinate origin is zb0.Due to Cb1、Cb2、Cb3Respectively it is in B1B2、
B3B4、B5B6The intersection point (as shown in Figure 5) of extended line, so Cb1、Cb2、Cb3Also in EbReference planes on.
According to definition above, three axial phasors of upper mounting plate are in E0Expression formula in coordinate system are as follows:
obxb=(xc2-xc3,yc2-yc3,zc2-zc3) (19)
obzb=obxb×obyb (21)
Define EbWith E0Between posture changing relationship: EbBy E0By the sequence successively rotatable coordinate axis counterclockwise of 3-1-2
(i.e. first along oz axis rotational angleOx ' axis rotational angle θ after rotation again, last oy " the axis rotational angle after rotation
γ),θ, γ are attitude angle.This posture cosine battle array rotated three times is respectively as follows:
E0To EbPosture changing cosine battle arrayAre as follows:
Six degree of freedom platform there are three the freedom degree of translational motion, three axis translation vector T ' of definition=[Δ x, Δ y,
Δ z] ', initial time EbWith E0The difference in height of coordinate origin is zb0, so the translation vector between Two coordinate system is corrected are as follows:
T=[Δ x Δ y Δ z+zbo]′ (24)
After platform carries out six-freedom motion, the coordinate transform battle array CT between Two coordinate system are as follows:
Any point B is in E on platformbCoordinate (x in coordinate systemb,yb,zb) after the translation of three axis and the rotation of three axis, it is right
Answer E0In corresponding coordinate (x, y, z) are as follows:
(x in above formulabo,ybo,zbo) it is EbCoordinate origin obIn E0In expression formula.
Due to the direction of translational motion not impact vector, EbThe axial vector of each reference axis is in E in coordinate system0In coordinate system
Expression formula are as follows:
(19)~(21) formula is compared into (27)~(29) formula, available triaxial attitude angleThe expression formula of θ, γ:
Three shaft position offsets are calculated below.By upper mounting plate origin obIn EbCoordinate (0,0,0) in coordinate system substitutes into formula
(26), it is available its in E0In coordinate (Δ x0, Δ y0, Δ z0+zbo), and because obForCenter, so three axis are inclined
Shifting amount are as follows:
Δxo=(xc1+xc2+xc3)/3
Δyo=(yc1+yc2+yc3)/3
Δzo=(zc1+zc2+zc3)/3-zb0 (30)
Triaxial attitude angle θ, γ,And Δ xo、Δyo、ΔzoI.e. the current posture of platform and position, Positive Solutions are complete
At.
The invention has the following advantages that utilizing the geometrical relationship of bearing tie point, crank, connecting rod on pedestal and upper mounting plate
Column write equation group, solve system of equation, and the triaxial attitude angle and relative position for obtaining upper mounting plate have obtained the six of brace and connecting rod framework
The posture normal solution method of freedom degree platform.
Detailed description of the invention
Fig. 1 is the six degree of freedom platform structure figure based on crank link mechanism；
Fig. 2 is platform base coordinate system schematic diagram proposed by the present invention；
Fig. 3 is each tie point schematic diagram of crank connecting link；
Fig. 4 is the initial phasor of crank and rotating phasor schematic diagram in reference planes；
Fig. 5 is each supporting point position coordinate schematic diagram of upper mounting plate；
Fig. 6 is connecting rod two end point and upper mounting plate long side extending line intersection point positional diagram；
Fig. 7 upper mounting plate coordinate system EbSchematic diagram.
Specific embodiment
The present invention will be further described with reference to the accompanying drawing, but protection scope of the present invention is not limited to following institute
It states.
Illustrate the embodiment of the present invention below.But embodiment below is only limitted to explain the present invention, protection model of the invention
Enclosing should include the full content of claim, and this hair can be thus achieved to person skilled in art by following embodiment
The full content of bright claim.
Below by taking certain type is based on the six degree of freedom platform of brace and connecting rod framework as an example, introduce the present invention and posture normal solution side
Method.
Six degree of freedom platform related parameter of certain type based on brace and connecting rod framework is as follows:
Crank connects the central point A of bearing with drive shaft on pedestali(xAi,yAi,zAi) (i=1 ..., 6) Ai(i=1 ...,
6) coordinate is sequentially are as follows:
With the crank length of motor driven axis connection | LAi|=0.12m, the length of connecting rod between crank and upper mounting plate | LB |
=0.47m.
Upper mounting plate is a hexagon, bond length hb=0.1m, long side length lb=0.7m, under original state, upper mounting plate six
A spring bearing center point is respectively: B1(-0.35,0.2598,0.3659)、B2(0.35,0.2598,0.3659)、B3(0.4,
0.1732,0.3659)、B4(0.05,-0.4330,0.3659)、B5(-0.05,-0.4330,0.3659)、B6(-0.4,
0.1732,0.3659)。
Push is completed according to the following steps:
(1) platform base coordinate system E is established0, six driving motors or its drive mechanism and song are determined according to mechanical structure
The position coordinates of handle junction:
A1(-0.21599,0.37413,0)、A2(0.21599,0.37413,0)、A3(0.432,0,0)、A4(0.216,-
0.374,0)、A5(-0.216,-0.374,0)、A6(-0.432,0,0)。
(2) according to crank current angle position, the bearing centre coordinate of crank Yu connecting rod junction is determined:
D1(0.1039,0.0600,0), D2(- 0.0943,0.0544, -0.0505), D3(0, -0.1088,0.0507), D4
(0.0943,0.0544, -0.0505), D5(- 0.0942,0.0544,0.0507), D6(0, -0.1089, -0.0505).
(3) according to the length of connecting rod, the geometrical relationship of each supporting point of upper mounting plate, one group of line of platform and supporting point is obtained
Property equation.According to the condition k=0.1111, L of this exampleB=0.47m, LC=0.9m.Obtain equation group:
(4) solve system of equation obtains posture and the position of upper mounting plate.
First seek C1(0.08132, -0.5132,0.3659), C2(0.4038,0.3270,0.3659), C3(- 0.4851,
0.1862,0.3659).
Triaxial attitude angle and three axle offset amounts are sought again:
θ=0 °, γ=9 °, Δ x0=0, Δ y0=0, Δ z0=0.
The above positive resolving Algorithm of posture is realized in same digitial controller.
Claims (5)
1. a kind of six free platform stance normal solution methods based on crank link mechanism, it is characterised in that: the following steps are included:
A. platform base coordinate system is established, the drive shaft and song of six driving motors or its transmission mechanism are determined according to mechanical structure
The center point coordinate of handle connection bearing；
B. according to crank current angle position, the bearing centre coordinate of crank Yu connecting rod junction is determined；
C. according to the length of connecting rod, the geometrical relationship of each supporting point of upper mounting plate, one group for obtaining platform and supporting point is linearly square
Journey；
D. solve system of equation obtains posture and the position of upper mounting plate.
2. a kind of six free platform stance normal solution methods based on crank link mechanism according to claim 1, the step A
The following steps are included:
(1) in platform base coordinate system E0In, using zenith direction as z-axis positive direction；With six axis driving motors or speed reducer rotary shaft
The reference planes that plane where line is z=0；Six crank connecting links and the supporting point of machine shaft junction constitute six sides
Shape, taking six crossing point of axes is coordinate origin o；With, for positive direction of the y-axis, right-hand direction is positive direction of the x-axis immediately ahead of platform；Around seat
The rotation angle of parameter ox axis is defined as pitching angle theta, and the rotation angle around oy axis is defined as roll angle γ, around the rotation angle of oz axis
Degree is defined as yaw angle
(2) defining crank and connecting the central point of bearing with the drive shaft of motor or its transmission mechanism is supporting point Ai(i=1 ...,
6), six supporting points constitute a hexagon A1A2A3A4A5A6；The coordinate A of six supporting points is determined according to mechanical structurei(xAi,
yAi,zAi) (i=1 ..., 6).
3. a kind of six free platform stance normal solution methods based on crank link mechanism according to claim 1, the step B
The following steps are included:
(1) definition and supporting point AiThe crank LA of connectioniIn coordinate system E0It is initial position when reference planes (xoy plane) are interior,
Remember LAiInitial position phasorRemember LAiCurrent time position With initial positionAngle αi, (i=1~
6, be positive counterclockwise)；
(2) remember the bearing centre point D of crank and connecting rod junctioni；It is coupled the center of the spring bearing of connecting rod and upper mounting plate junction
It puts and isCrank LAiRotary shaft be vector
(3) crank initial positionAnd rotating vector piIt is distributed in reference planes, wherein pi=[xAi,yAi,zAi]；
In being gained knowledge according to movement, any vector surrounds piRotate angle [alpha]iSpin matrix expression formula shift onto it is available: crank
LAiPostrotational vectorExpression formula are as follows:And then available DiCoordinate (the x of pointdi,ydi,zdi):
4. a kind of six free platform stance normal solution methods based on crank link mechanism according to claim 1, the step C
The following steps are included:
(1) each spring bearing centre of sphere central point of upper mounting plate is defined are as follows:BiIn coordinate system E0In coordinate beBiAltogether there are six point, this six points constitute hexagon B1B2B3B4B5B6；
(2) remember hexagon long side length | B1B2|=| B3B4|=| B5B6|=lb, bond length | B2B3|=| B4B5|=| B6B1|=
hb, the extended line of three long sides meets at three point Cb1、Cb2、Cb3, these three points are in coordinate system E0In coordinate be (xcj,ycj,zcj)(j
=1,2,3)；For an equilateral triangle；
(3) triangleSide CbjCbkThe length of (j=1,2,3k=1,2,3 and j ≠ k) are as follows: | CbjCbk|=2hb+lb,
Bi、DiWith adjacent CbjMay be constructed a triangle, if Bi、DiBetween connecting rod be LBi, vector expression isIt is long
Degree is LB, then according to Bi、Di、CbjCorrelation it is available:
(4) because | CbjBi|=hbIf k=hb/CbjCbk|=hb/2hb+lb,
It is available:
And then it obtains about Cb1、Cb2、Cb3Three coordinate (xcj,ycj,zcj) (j=1,2,3) six equations；
Further according to | CbjCbk|=LC(j=1,2,3, k=1,2,3 and j ≠ k), can arrange and write triangleSide length it is public
Formula is obtained about (xcj,ycj,zcj) (j=1,2,3) the other three equation；Above-mentioned nine equations, constitute about xc1、yc1、
zc1、xc2、yc2、zc2、xc3、yc3、zc3The equation group of this nine unknown numbers.
5. a kind of six free platform stance normal solution methods based on crank link mechanism according to claim 1, the step D
The following steps are included:
(1) equation group constituted using nine equations in the method solution procedure C for solving Nonlinear System of Equations, it is available
xc1、yc1、zc1、xc2、yc2、zc2、xc3、yc3、zc3This nine unknown numbers；
(2) the upper mounting plate coordinate system E connected firmly with upper mounting plate is definedb: EbReference planes (zb=0 plane) take six branch of upper mounting plate
Support bearing center point Bi(i=1 ..., 6) place plane；Origin obIt is taken as upper mounting plate geometric center；
(3) remember that six cranks are in a horizontal position (i.e. αi=0, i=1,2 ..., 6) upper mounting plate is in original state, initial shape when
O under statebxbAxis and platform base coordinate system E0Middle ox axis is parallel；obybAxis and platform base coordinate system E0Middle oy axis is parallel；At this point,
EbCoordinate origin distance E0The height of coordinate origin is zb0；Three axial phasors of upper mounting plate are in E0Table in coordinate system
Up to formula are as follows:
obxb=(xc2-xc3,yc2-yc3,zc2-zc3)
obzb=obxb×obyb
(4) E is definedbWith E0Between posture changing relationship: EbBy E0By the sequence of 3-1-2, successively rotatable coordinate axis counterclockwise is (i.e.
First along oz axis rotational angleOx ' axis rotational angle θ after rotation again, last oy " the axis rotational angle γ after rotation),θ, γ are attitude angle；The posture cosine battle array rotated three times is respectively as follows:
E0To EbPosture changing cosine battle arrayAre as follows:
(5) there are three the freedom degree of translational motion, three axis translation vector T '=[Δ x, Δ y, Δs of definition for six degree of freedom platform
Z] ', initial time EbWith E0The difference in height of coordinate origin is zb0, so the translation vector between Two coordinate system is corrected are as follows:
T=[Δ x Δ y Δ z+zbo]′
After platform carries out six-freedom motion, the coordinate transform battle array CT between Two coordinate system are as follows:
Any point B is in E on platformbCoordinate (x in coordinate systemb,yb,zb) after the translation of three axis and the rotation of three axis, corresponding E0
In corresponding coordinate (x, y, z) are as follows:(x in above formulabo,ybo,zbo) it is EbCoordinate origin obIn E0
In expression formula；
(6) due to the direction of translational motion not impact vector, E is obtainedbThe axial vector of each reference axis is in E in coordinate system0Coordinate system
In expression formula are as follows:
By three expression formulas and the comparison of above three expression formula in (3), available triaxial attitude angleThe expression formula of θ, γ:
(7) by upper mounting plate origin obIn EbCoordinate (0,0,0) in coordinate system substitutes into formula (26), it is available its in E0In
Coordinate (Δ x0, Δ y0, Δ z0+zbo), and because obForCenter, acquire triaxial attitude angle θ, γ,And Δ xo、Δ
yo、ΔzoThat is the current posture of platform and position:
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Cited By (4)
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