Embodiment
Below in conjunction with accompanying drawing and example, the present invention is described in further detail.
The present invention provides the parallel institution that a kind of new two translation one rotates, i.e. 2-PSS&1-PPR parallel institutions, such as Fig. 1
Shown, described parallel institution has along x, y-axis translation and the three degree of freedom rotated along x-axis.Described parallel institution is by moving
Platform, silent flatform and 3 side chain compositions.The moving platform is by A1、A2、A3The triangle projective planum of three point compositions, and A1A2
Length is equal to A1A3Length;Described silent flatform is by b1、b2、b3The U-shaped right angle frame plane of three line slideway compositions, its
In, guide rail b2With guide rail b3Along x-axis, guide rail b1Perpendicular to x-axis;Three side chains are respectively A1B1、A2B2、A3B3。
Side chain A1B1It is divided into two sections, is connected between epimere and moving platform by revolute pair, revolute pair axis is along x-axis side
To;Connected between epimere and hypomere by prismatic pair, prismatic pair is along the x-axis direction;Pass through mobile parafacies between hypomere and silent flatform
Even, prismatic pair is along the y-axis direction.Side chain A2B2、A3B3Structure is identical, and upper end is connected by ball pair with moving platform, and lower end passes through ball pair
With slide block B2, slide block B3It is connected, slide block B2, slide block B3It is located at guide rail b respectively2, guide rail b3On, guide rail b2, guide rail b3Along x-axis side
To.Positioned at three guide rail b1、b2、b3On three slide block Bs1、B2、B3For the driving link of mechanism.
The free degree and its property of the mechanism can be easily analyzed using spinor method.Now analyze pact of each side chain to moving platform
Beam, it is clear that side chain A1B1There are three constraints to moving platform, power and the torque around y, z direction respectively in the z-direction.Focus on branch
Chain A2B2And A3B3To constraint caused by moving platform.As side chain A2B2And A3B3During with x-axis out of plumb, side chain A2B2Or A3B3In include
Seven kinematic pair spinors public reciprocity spinor is not present, now side chain A2B2And A3B3To moving platform without constraint, i.e., moving platform this
When have along x, y-axis translation and along x-axis rotate three degree of freedom.
The position forecast of parallel institution refers to solve the carry-out bit of moving platform according to the input parameter of each drive pair of mechanism
Appearance parameter, and Kinematics analysis refers to according to the pose parameter of moving platform come the input parameter of each drive pair of reverse.For described three
Freedom degree parallel connection mechanism, input parameter are position of 3 sliding blocks in respective guide rail in driving link.
1.1 position equation
Establish the coordinate system shown in Fig. 1, wherein OAxAyAzATo be fixed on the moving coordinate system on moving platform, OBxByBzBIt is solid
The quiet coordinate system being scheduled on silent flatform, moving platform A1A2A3For isosceles triangle, whereinTriangle base width is
2b, a height of a, guide rail b2With guide rail b3The distance between be 2c, revolute pair A1Axis and slide block B2、B3The vertical range of upper sphere pair is
H, side chain A2B2And A3B3Bar length be l.
The input parameter of mechanism is:P=(y1,x2,x3)T, wherein y1、x2、x3Respectively slide block B1、B2、B3In quiet coordinate system
In y, x coordinate.The output parameter of mechanism is q=(x, y, α)T, wherein x, y is moving coordinate system origin OAIn quiet coordinate system
X, y-coordinate, α are the anglec of rotation of the relatively quiet coordinate system of moving coordinate system around x-axis.
A1、A2、A3Coordinate in moving coordinate system is:(0,0,0)T、(a,-b,0)T、(a,b,0)T, i.e.,:
B1、B2、B3Coordinate in quiet coordinate system is:(x1,0,0)T、(x2,-c,0)T、(x3,c,0)T, i.e.,:
Moving coordinate system origin OAIt is expressed as under quiet coordinate system:
P=(x, y, h)T (3)
Then A1、A2、A3It is represented by under quiet coordinate system:
Wherein R is spin matrix, it is known that moving platform is α around x-axis corner, then:
Formula (1), formula (3), formula (5) are substituted into formula (4) and can obtain:
Mechanism has following constraints:
Formula (2) and formula (6) are substituted into formula (7) position equation of mechanism can be obtained and be:
1.2 Kinematics analysis
For the position equation of mechanism, solved equation using mechanism input parameter as unknown number can obtain mechanism position it is anti-
Solution.It might as well set:
It is apparent from r2、r3Side chain A in as Fig. 12B2、A3B3In yBOBzBThe projection of plane.
Work as l2-ri 2> 0 is riDuring < l (i=2,3), it can be obtained by mechanism position equation:
The formula of the above three is the Kinematics analysis of mechanism.Wherein x2And x3Expression formula in contain sign, positive sign represents to slide
Block is in the right side (x-axis is positive) of moving platform ball pivot, and negative sign represents that sliding block is in the left side (x-axis negative sense) of moving platform ball pivot.Machine
Four groups of Kinematics analysis of structure are as shown in Figure 2.
1.3 position forecast
Arranged by parallel institution position equation (8):
Wherein:
Arrange,
Wherein:
Further abbreviation obtains:
I.e.:
Wherein:
Again by sin2x+cos2X=1 obtains the unary biquadratic equation on x:
E4x4+E3x3+E2x2+E1x+E0=0 (15)
Wherein:
Thus Positive Solutions are converted for the Solve problems of unary biquadratic equation.Because unary biquadratic equation is up to
Four real solutions, therefore at most four groups of analytic solutions be present in the 3-freedom parallel mechanism.
2.1 rate equation
The first derivative of position equation (8) both sides seeking time is obtained:
Wherein:For moving platform output speed,For mechanism input speed,
When | Jp| when ≠ 0,Now haveWherein:
2.2 statics equations
The static problems of 2-PSS&1-PPR parallel institutions are solved using the principle of virtual displacement.The principle of virtual displacement is also known as virtual work
Principle, can be stated as by preferable, two-sided, scleronomic constraint, steady constraint system of material points keep balance NSC be:All effects
The virtual work sum that active force on system of material points is made to the virtual displacement of its application point is zero.If the driving force of three driving sliding blocks
For f=(f1 f2 f3)T, moving platform power output and torque are F=(Fx Fy Mx)TIfFor mechanism force Jacobian matrix, then
Have:
Obtained according to the principle of virtual work:
Equation (21) both sides divided by δ t can be obtained:
Obtained by above formula:Wherein J is mechanism Jacobian matrix, i.e. J=Jq -1Jp, therefore:
3 kinematics of mechanism emulate
Utilize virtual prototype software analysis institution motion process, model as shown in Figure 3, in three freedom of mechanism moving platform
Apply following motion on degree respectively:X=0.75t2, y=0.75t2, α=0.01t2.Run duration is arranged to 6s.Counted by emulation
The speed-time curve that calculation obtains three sliding blocks is as shown in Figure 4.
Fig. 5 is according to mechanism Jacobian matrix, with the speed-time curve of Matlab three sliding blocks being calculated.It is logical
Cross the contrast with Fig. 4 and can be seen that the two is nearly identical, this is just demonstrating Jacobian matrix that theory deduction obtains just
True property.
Singularity is the build-in attribute of parallel institution, and its performance to parallel institution has a major impact.At parallel institution
When Singularity, mechanism will appear from uncontrollable extra dof, or lose some frees degree, the power transmission performance of mechanism
Also what can be become is very poor, and now parallel institution can not use.Therefore mechanism should be avoided to be in Singularity in design and application.Machine
The unusual rule of mechanism is mainly disclosed in structure theory by the Jacobian matrix of research institution.
When speed, which transmits Jacobian matrix, meets following condition:
|Jp|=0 (24)
Mechanism is in first kind Singularity.Now the anti-solution of the speed of mechanism is not present, and no matter ram speed is much, moves
Platform speed is always zero, and mechanism output campaign function is lost;No matter by more heavy loads, sliding block is also not required to appoint moving platform simultaneously
What driving force goes to balance, mechanism rigidifying.Above formula is deployed:
|Jp|=(x2-x-a)(x3- x-a)=0 (25)
If above formula is set up, x2- x-a=0 or x3- x-a=0, according to the anti-solution formula of mechanism, now there is l2-r2 2=0 or l2-
r3 2=0, the anti-solution of mechanism is unique.Now mechanism side chain A2B2Or side chain A3B3Respectively with guide rail b2And b3Vertically, mechanism moving platform
Reach the border of its working space.
Fig. 6 is b=45mm, c=90mm, h=40mm, l during l=100mm2-r2 2And l2-r3 2The distribution map of value.Scheming
Middle l2-r2 2=0 and l2-r3 2At=0 isopleth, mechanism reaches first kind Singularity.
When speed, which transmits Jacobian matrix, meets following condition:
|Jp| ≠ 0, | Jq|=0 (26)
Mechanism is in the second class Singularity.Now the speed normal solution of mechanism is not present, and mechanism moving platform appearance can not
The extra dof of control, while mechanism can not realize the balance of power when sliding block bears load.
Formula (26) is deployed:
|Jq|=b (x+a-x3)(ysinα+csinα-hcosα)-b(x+a-x2)(-ysinα+csinα+hcosα)
Arrange the unusual occurrence condition of the second class is:
|Jq|=b (x2+x3-2x-2a)(hcosα-ysinα)+bc(x2-x3) sin α=0 (27)
According to the anti-solution formula of mechanism position, solved when mechanism is counter in the presence of i.e. l2-ri 2Have when >=0 (i=2,3):
Wherein,
It above formula substituted into formula (27) can obtain pose parameter and only include mechanism output variable y, the unusual generation of α the second class
Condition:
Fig. 7 is b=45mm, when c=90mm, h=40mm | Jq| the distribution map of value.Four small figures correspond to l=respectively
100mm, l=150mm, l=200mm, l=300mm.In figure | Jq|=0 mechanism of isopleth (black dotted lines) place reaches second
Class Singularity.It was found from above-mentioned analysis, the unusual working space of the more big non-first kind of l is bigger, but l increase is for expanding
The effect of non-second class singular space is little.