CN102023643A - 3-PRP planar three-degree-of-freedom parallel mechanism positioning method - Google Patents

Abstract

The invention provides a 3-PRP planar three-degree-of-freedom parallel mechanism positioning method, which comprises the following steps of: establishing a moving coordinate system which moves along with a moving component, and determining a moving principle; acquiring a zero state parameter needed by positioning through a debugging method; determining an initial state during inverse solution of the position; and finally obtaining driving quantity needed by platform movement through an inverse solution method according to coordinates of two position points before and after movement. A relationship established between the moving coordinate system and a ground static coordinate system of a measuring instrument is simple and clear and is easy to calculate; the zero state parameter obtained through the debugging method has higher accuracy than that obtained through other measuring methods; the inverse solution method provides method support in the whole moving process of the mechanism and can meet the requirement in industrial actual application; and the positioning problem is solved to push the application of the 3-PRP planar three-degree-of-freedom parallel mechanism; therefore, the mechanism can accurately reach the expected position in a plane and can be applied to the industrial fields such as accurate alignment and the like.

Description

3-PRP planar three freedom parallel institution localization method

Technical field

The invention belongs to mechanical location control field, relate to the parallel institution position analysis, be specifically related to a kind of planar three freedom parallel institution of 3-PRP structure.

Background technology

Parallel institution has that the rigidity of structure is big, load-bearing capacity is high, kinematic accuracy is good and the position is counter separates simple and be easy to the unexistent advantage of many polyphone robots such as force feedback control, has closely become one of the main focus in domestic and international robot research field during the last ten years.Domesticly a large amount of deep researchs have been done at aspects such as the structure of parallel institution, kinematics, dynamics, stated accuracies such as colleges and universities such as Tsing-Hua University, Harbin Institute of Technology, University On The Mountain Of Swallows, Beijing Institute of Technology, Shenyang robotization research institute of the Chinese Academy of Sciences and scientific research institution.

What the present invention is directed to is a kind of planar three freedom parallel institution of 3-PRP structure, this kind mechanism is made up of movable platform and three active branched chain, the rectangle movable platform is driven by active branched chain, carrying out the plane with respect to fixed pan moves and rotates, each active branched chain is made up of an active slip pair (P), a revolute pair (R) and a passive sliding pair (P), and the active slip pair is connected with stationary platform, passive sliding pair is connected with free cheek.When active slip pair moving linearly, by revolute pair and passive sliding pair free cheek is exerted one's influence, cause moving or rotating of it.This parallel institution has advantages such as rapidity, high precision, high rigidity, load-bearing capacity be big.Yet, the characteristics because each parts of parallel institution are coupled mutually, its difficulty that realizes the location is more much bigger than serial mechanism.Domestic research and the application that does not also have this kind 3-PRP mechanism, this kind mechanism does not still have the localization method solution of announcement at present.

The location of mechanism relates to that coordinate mechanism kinematic mode, coordinate are set up, kinematical equation simultaneous and knowledge such as find the solution, and is intended to make mechanism accurately to navigate to the position of expectation, and whether estimate the location by the gauge point in the mechanism usually accurate.Orientation problem to be solved refers to how to make the position of mechanism by mechanism kinematic arrival expectation.This problem also can be summed up as: " so that mechanism arrives the position of expectation, how driving this mechanism for the motion that realizes expecting? " localization method is the method that orientation problem adopted that solves.

The basic skills that solves the parallel institution orientation problem has that position normal solution and position are counter separates two kinds of analytical approachs.The position normal solution refers under the situation of given driving link position, finds the solution the process of other driven member position shape attitude according to the mechanism position system of equations; Counter the separating in position is meant under given mechanism driven member position shape attitude, finds the solution the process of driving link locus according to the mechanism position system of equations.By to the anti-analysis of separating of certain mechanism position normal solution and position, again their applied in any combination just can be obtained localization method at this kind mechanism, realize its accurate location in the space, for road has been paved in the practical application of mechanism.The position is counter for parallel institution separates easylier than position normal solution, and purposes is wider.At present, as be widely used in the Stewart six-degree-of-freedom parallel connection mechanism of electromechanical equipments such as aircraft simulation, turntable, just made full use of the transformational relation between moving coordinate system and the quiet coordinate system, obtained that the position is counter separates, realize its location in the space.

Summary of the invention

The objective of the invention is orientation problem, propose a kind of 3-PRP planar three freedom parallel institution localization method for the planar three freedom parallel institution that solves the 3-PRP structure.

A kind of 3-PRP planar three freedom of the present invention parallel institution localization method specifically may further comprise the steps:

Step 1, on the plane at place, three revolute pair centers, set up a plane moving coordinate system that together moves with motion platform, the component of described three the some change in coordinate axis direction of revolute pair center in this moving coordinate system is always zero; Wherein, in this moving coordinate system, the coordinate axis at the place, revolute pair center of first active branched chain is the y axle, and the coordinate axis at the place, revolute pair center of second and the 3rd active branched chain is the x axle.

Step 2, obtain machine initial point M by following debugging step ₀Under four kinds of states,, and write down these four marks of sitting quietly with respect to the mark of sitting quietly of quiet coordinate system; Described machine initial point M ₀It is the gauge point that is provided with in the center of rectangle movable platform;

Step 2.1, make the secondary all linear drives distance A of active slip of three active branched chain, obtain machine initial point M ₀Mark (the mX that sits quietly of this moment ₁, mY ₁);

Step 2.2, make second active branched chain the active slip pair again linear drives obtain machine initial point M apart from B ₀Sitting quietly of this moment is designated as (mX ₂, mY ₂);

Step 2.3, make first active branched chain the active slip pair again linear drives obtain machine initial point M apart from B ₀Sitting quietly of this moment is designated as (mX ₃, mY ₃);

Step 2.4, make second active branched chain the active slip pair again linear drives obtain machine initial point M apart from B ₀Sitting quietly of this moment is designated as (mX ₄, mY ₄).

Step 3, according to four that obtain in the step 2 marks of sitting quietly, according to the relation of coordinate system transformation, obtain the information of zero-bit state:

${\mathrm{\θ}}_1=\arctan \frac{{\mathrm{mY}}_2-{\mathrm{mY}}_3}{{\mathrm{mX}}_3-{\mathrm{mX}}_2}$

${\mathrm{\θ}}_2=2\arctan \frac{{\mathrm{mY}}_2-{\mathrm{mY}}_3}{{\mathrm{mX}}_3-{\mathrm{mX}}_2}$

${\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">X</figure-callout>}}_1=\frac{({\mathrm{mY}}_4-{\mathrm{mY}}_2)\sin {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2+(A+B-{\mathrm{mX}}_4)\sin {\mathrm{\&theta;}}_1\cos {\mathrm{\&theta;}}_2+(m{X}_2-A)\cos {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2-({\mathrm{mX}}_1-A)(\sin {\mathrm{\&theta;}}_2-\sin {\mathrm{\&theta;}}_1)}{\sin ({\mathrm{\&theta;}}_2-{\mathrm{\&theta;}}_1)-(\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}$

${\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1=\frac{({\mathrm{mX}}_2-X_1-A)\cos {\mathrm{\&theta;}}_1+(X_1+A+B-{\mathrm{mX}}_4)\cos {\mathrm{\&theta;}}_2-{\mathrm{mY}}_2\sin {\mathrm{\&theta;}}_1+{\mathrm{mY}}_4\sin {\mathrm{\&theta;}}_2}{\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1}$

$X_2=\frac{(Y_2+A+B+{\mathrm{mY}}_2)\cos {\mathrm{\&theta;}}_1+(-Y_2-A-2B+{\mathrm{mY}}_4)\cos {\mathrm{\&theta;}}_2-{\mathrm{mX}}_2\sin {\mathrm{\&theta;}}_1+{\mathrm{mX}}_4\sin {\mathrm{\&theta;}}_2}{\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1}$

$Y_2=\frac{({\mathrm{mX}}_2-{\mathrm{mX}}_4){\sin \mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2+(A+2B-{\mathrm{mY}}_4)\sin {\mathrm{\&theta;}}_1\cos {\mathrm{\&theta;}}_2+({\mathrm{mY}}_2-A-B)\cos {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2-({\mathrm{mY}}_1-A)({\sin \mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}{\sin ({\mathrm{\&theta;}}_2-{\mathrm{\&theta;}}_1)-(\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}$

Y ₃＝Y ₂

$X_3=X_2+\frac{B}{\tan {\mathrm{\&theta;}}_1}$

Wherein, θ ₁For the moving coordinate system that forms in the step 2.2 with respect to the moving coordinate system that forms in the step 2.1; The angle that is rotated counterclockwise; θ ₂The angle that is rotated counterclockwise with respect to the moving coordinate system that forms in the step 2.1 for the moving coordinate system that forms in the step 2.4; (X ₁, Y ₁), (X ₂, Y ₂) and (X ₃, Y ₃) be respectively the coordinate of three revolute pair centers with respect to quiet coordinate system;

The information of step 4, the zero-bit state that obtains according to step 3 is demarcated original state, and described original state is meant the state after motion that be set by the user or last stops; The original state that demarcation obtains is:

$\mathrm{\&theta;}=\arctan \frac{Y_2-Y_3+b-c}{X_3-X_2}$

P＝(Y ₂+b-Y ₁)sinθcosθ+X ₂sin ²θ+(X ₁+a)cos ²θ

Q＝-(X ₁+a-X ₂)sinθcosθ+Y ₁sin ²θ+(Y ₂+b)cos ²θ

Wherein, three active branched chain were with respect to the linear advancement amount under the zero-bit state when a, b, c were respectively the original state that the user sets, θ is the angle that original state counterclockwise moves during with respect to the zero-bit state, P is the distance that original state moves at the x direction of principal axis during with respect to the zero-bit state, and Q is the distance that original state moves at the y direction of principal axis during with respect to the zero-bit state;

Step 5, according to two location points; Before and after the movable platform motion,, obtain with respect to three straight line active drive amounts under the original state with respect to the mark of sitting quietly of quiet coordinate system:

$\mathrm{\&Delta;}{\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">X</figure-callout>}}_1=\frac{{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1\sin (\mathrm{\&theta;}+r)-S\sin (\mathrm{\&theta;}+r)+R\cos (\mathrm{\&theta;}+r)}{\cos (\mathrm{\&theta;}+r)}-X_1-a,$

$\mathrm{\&Delta;}{Y}_2=\frac{-X_2\sin (\mathrm{\&theta;}+r)+R\sin (\mathrm{\&theta;}+r)+S\cos (\mathrm{\&theta;}+r)}{\cos (\mathrm{\&theta;}+r)}-Y_2-b,$

$\mathrm{\&Delta;}{Y}_3=\frac{-X_3\sin (\mathrm{\&theta;}+r)+R\sin (\mathrm{\&theta;}+r)+S\cos (\mathrm{\&theta;}+r)}{\cos (\mathrm{\&theta;}+r)}-Y_3-c,$

Wherein, two location points are meant two the some M and the N of on movable platform mark, and the sitting quietly mark of some M before and after the movable platform motion is respectively (Mx ₁, My ₁) and (Mx ₂, My ₂), the sit quietly mark of some N before and after the movable platform motion is respectively (Nx ₁, Ny ₁) and (Nx ₂, Ny ₂), r is the angle that counterclockwise moves when moving coordinate system is with respect to the zero-bit state after the motion,

R is the distance that moves at the x direction of principal axis when moving coordinate system is with respect to the zero-bit state after the motion, and R=Gcos (θ+r)+Hsin (θ+r); S is the distance that moves at the y direction of principal axis when moving coordinate system is with respect to the zero-bit state after the motion, and S=-Gsin (θ+r)+Hcos (θ+r); Wherein, parameter G, H are respectively:

G＝-Mx ₁cosθ+My ₁sinθ-Qsinθ+Pcosθ+Mx ₂cos(θ+r)-My ₂sin(θ+r)，

H＝Mx ₁sinθ-My ₁cosθ+Psinθ+Qcosθ+Mx ₂sin(θ+r)+My ₂cos(θ+r)，

According to above-mentioned three straight line active drive amounts that obtain, the controlling and driving motor is exported corresponding displacement, promotes movable platform and realizes accurately location.

Advantage of the present invention and good effect are:

(1) use moving coordinate system, simple and clear with the quiet coordinate system relation in ground at surveying instrument (it is first-class to make a video recording) place, be easy to calculate;

(2) the zero-bit state parameter that obtains of adjustment method is than the higher precision that has that obtains with other measurement means, and at machine timing signal is installed and can be recorded;

(3) instead separate the step that method is used based on reality, promptly by the zero-bit state to original state, the overall process of state provides the method support to moving afterwards again, can satisfy the demand in the industrial practical application;

(4) solution of orientation problem can be played the effect of propelling to the application of 3-PRP planar three freedom parallel institution, makes this mechanism can accurately arrive the desired locations in the plane, can be applicable to accurately industrial circle such as alignings grade.

Description of drawings

Fig. 1 is a 3-PRP planar three freedom parallel institution structural representation;

Fig. 2 is a 3-PRP planar three freedom parallel institution motion synoptic diagram;

Fig. 3 is the flow chart of steps of localization method of the present invention;

Fig. 4 is the conversion synoptic diagram of the moving coordinate system set up in the step 1 of the present invention;

Fig. 5 a be in the step 1 of the present invention moving coordinate system set up and adjustment method step 2.1 in the synoptic diagram of parallel institution motion state;

Fig. 5 b is the synoptic diagram of parallel institution motion state in the adjustment method step 2.2 of the present invention;

Fig. 5 c is the synoptic diagram of parallel institution motion state in the adjustment method step 2.3 of the present invention;

Fig. 5 d is the synoptic diagram of parallel institution motion state in the adjustment method step 2.4 of the present invention;

To be that the position is counter in the localization method step 5 of the present invention separate used moving coordinate system synoptic diagram to Fig. 6.

Embodiment

The present invention is described in further detail below in conjunction with drawings and Examples.

Be illustrated in figure 1 as 3-PRP planar three freedom parallel institution structural representation, a movable platform Table is arranged on fixed pan, Table is connected with fixed pan by three active branched chain, each active branched chain all is made up of active slip pair, turning joint, passive sliding pair, and turning joint is exactly a revolute pair.First active branched chain is made up of secondary motor1 of active slip, revolute pair R1 and passive sliding pair P1; Second active branched chain is made up of secondary motor2 of active slip, revolute pair R2 and passive sliding pair P2; The 3rd active branched chain is made up of secondary motor3 of active slip, revolute pair R3 and passive sliding pair P3.The synoptic diagram that Fig. 2 moves in the space for 3-PRP planar three freedom parallel institution, as can be seen, by the linear drives of active slip pair in three active branched chain, movable platform can realize the motion of two one-movement-freedom-degrees and a rotational freedom in the plane.The present invention is a kind of 3-PRP planar three freedom parallel institution localization method, specifically as shown in Figure 3, comprises five steps:

Step 1, set up a moving coordinate system that together moves with moving component.

The foundation of moving coordinate system is the first step that solves orientation problem, sets up a coordinate system that together moves with moving component, is used to stipulate the coordinate of any location point on these parts.Moving component refers to movable platform herein.

At first introduce the present invention based on transformation matrix of coordinates thought.Coordinate transform is basic geometric knowledge, at this, expresses the transformational relation of coordinate between two coordinate systems with transformation matrix of coordinates.As Fig. 4, coordinate system Xp ₀Op ₀Yp ₀With coordinate system Xp ₁Op ₁Yp ₁Following relation is arranged: coordinate system Xp ₀Op ₀Yp ₀Be rotated counterclockwise angle θ, obtain coordinate system Xp ₁Op ₁Yp ₁, coordinate Xp ₁Op ₁Yp ₁With respect to coordinate system Xp ₀Op ₀Yp ₀Along directions X displacement P, Y direction displacement Q.If 1 N is respectively with respect to the coordinate of two coordinate systems that (x is y) with (x in the plane _p, y _p).So, can get according to coordinate transform:

$\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=\left[\begin{array}{ccc}\cos \mathrm{\&theta;}& \sin \mathrm{\&theta;}& P\\ -\sin \mathrm{\&theta;}& \cos \mathrm{\&theta;}& \mathrm{<figure-callout\; id="0"\; label="Q"\; filenames="HSA00000290529300031.png"\; state="\{\{state\}\}">Q</figure-callout>}\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_p\\ y_p\\ 1\end{array}\right]---\left(1.1\right)$

Write equation (1.1) as following form, T ₀Be the coordinate system transformation matrix, represented in the plane a bit before motion transformation of coordinates relation in coordinate system and the motion back coordinate system.

$\left[\begin{array}{c}{x}_{\mathrm{<figure-callout\; id="1"\; label="p\; y\; p"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">p</figure-callout>}}& \\ y_p{}_{}\\ 1\end{array}\right]={\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HSA00000290529300031.png"\; state="\{\{state\}\}">T</figure-callout>}}_0\&CenterDot;\left[\begin{array}{c}\mathrm{<figure-callout\; id="1"\; label="x\; y"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">x</figure-callout>}& \\ y\\ 1\end{array}\right],$ If ${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HSA00000290529300031.png"\; state="\{\{state\}\}">T</figure-callout>}}_0=\left[\begin{array}{ccc}\cos \mathrm{\&theta;}& -\sin \mathrm{\&theta;}& Q\sin \mathrm{\&theta;}-P\cos \mathrm{\&theta;}\\ \sin \mathrm{\&theta;}& \cos \mathrm{\&theta;}& -P\sin \mathrm{\&theta;}-Q\cos \mathrm{\&theta;}\\ 0& 0& 1\end{array}\right]---\left(1.2\right)$

Moving coordinate system be based upon on the movable platform, with the coordinate system that movable platform is moved, be different from the quiet coordinate system that is fixed in ground.Can either bring the facility of setting coordinate by the foundation of moving coordinate system, can provide obvious natural equation again, for position analysis provides system of equations outside the coordinate transform.Be not difficult to obtain according to mechanical mechanism: no matter under which kind of motion state, as shown in Figure 2, the center of the revolute pair R1 of 3-PRP parallel institution, R2 and R3 is constant to the distance of movable platform, and the shape of movable platform is a rectangle.If establish coordinate systems with three revolute pair centers, then this coordinate system is with the motion campaign of rectangle movable platform, and is all constant with respect to movable platform in any motion.Revolute pair is in the intermediate link between driving and the movable platform simultaneously, then can utilize this communication link that the moving axes and the mark of sitting quietly are connected.

So, shown in Fig. 5 a, set up moving coordinate system with three revolute pair centers, if rectangle movable platform motion, then moving coordinate system still keeps relative static with this platform.On the movable platform any is before and after motion, and its moving axes with respect to moving coordinate system is constant.

Shown in Fig. 5 a, quiet coordinate system described in the embodiment of the invention, be fixed in ground, in the same plane with the rectangle movable platform, set up on their own by the user as the case may be, second active branched chain and the 3rd active branched chain are positioned on the x direction of principal axis of this quiet coordinate system, perhaps are parallel on the x direction of principal axis of this quiet coordinate system.The moving coordinate system of being set up, the coordinate axis at the place, revolute pair center of first active branched chain is the y axle, the coordinate axis at the place, revolute pair center of second and the 3rd active branched chain is the x axle.When the 3-PRP parallel institution was in zero-bit state shown in Fig. 5 a, quiet coordinate system was parallel with moving coordinate system.

In the motion process of parallel institution, on the moving axes axle, promptly these three points are always zero at the component of a change in coordinate axis direction of plane moving coordinate system all the time at the revolute pair center of three active branched chain.Three coordinates of revolute pair center in moving coordinate system be respectively (0, Yp ₁), (Xp ₂, 0) and (Xp ₃, 0), can write out the kinematical equation of three active branched chain revolute pair centers by equation (1.2), also be that moving coordinate system is set up the natural equation that brings:

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="Xp"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Xp</figure-callout>}}_1=({\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">X</figure-callout>}}_1+{\mathrm{\&Delta;X}}_1)\cos \mathrm{\&theta;}-{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1\sin \mathrm{\&theta;}+Q\sin \mathrm{\&theta;}-P\cos \mathrm{\&theta;}=0\\ {\mathrm{Yp}}_2=X_2\sin \mathrm{\&theta;}+(Y_2+{\mathrm{\&Delta;Y}}_2)\cos \mathrm{\&theta;}-P\sin \mathrm{\&theta;}-Q\cos \mathrm{\&theta;}=0\\ {\mathrm{Yp}}_3=X_3\sin \mathrm{\&theta;}+(Y_3+{\mathrm{\&Delta;Y}}_3)\cos \mathrm{\&theta;}-P\sin \mathrm{\&theta;}-Q\cos \mathrm{\&theta;}=0\end{array}\right.---\left(1.3\right)$

Wherein, (Xp ₁, Yp ₁), (Xp ₂, Yp ₂) and (Xp ₂, Yp ₂) be respectively three coordinates of revolute pair center in moving coordinate system; (X ₁, Y ₁), (X ₂, Y ₂) and (X ₃, Y ₃) be respectively three initial positions of revolute pair center in quiet coordinate system; Δ X ₁, Δ Y ₂With Δ Y ₃Be respectively the push-in stroke of three active branched chain linear drives.

Step 2, obtain the coordinate under the one of four states of machine initial point by adjustment method.

Obtaining the required zero-bit state parameter of localization method information is the prerequisite that solves orientation problem, and these parameters comprise: be under the zero situation at junior three linear drives push-in stroke, three revolute pair centers are with respect to the coordinate (X of quiet coordinate system ₁, Y ₁), (X ₂, Y ₂) and (X ₃, Y ₃).And these parameters are difficult to obtain exact value by measurement, and this just needs adjustment method to obtain them.When machine is installed timing signal, measure the former dot information of machine that obtains mark by four steps of adjustment method, try to achieve the zero-bit state parameter that solves orientation problem according to these information again.

Described machine initial point is meant the gauge point M that the machine timing signal is provided with in the center of rectangle movable platform ₀It all is to take pictures and carry out Flame Image Process by the camera of looking down from the movable platform top to obtain in actual use that all targets of sitting quietly obtain.

Four steps of adjustment method are:

Step 2.1, make the secondary all linear drives distance A of active slip of three active branched chain, obtain machine initial point M ₀Mark (the mX that sits quietly of this moment ₁, mY ₁);

Step 2.2, make second active branched chain the active slip pair again linear drives obtain machine initial point M apart from B ₀Sitting quietly of this moment is designated as (mX ₂, mY ₂);

Step 2.3, make first active branched chain the active slip pair again linear drives obtain machine initial point M apart from B ₀Sitting quietly of this moment is designated as (mX ₃, mY ₃);

Step 2.4, make second active branched chain the active slip pair again linear drives obtain machine initial point M apart from B ₀Sitting quietly of this moment is designated as (mX ₄, mY ₄).

Step 3, obtain zero message.

Find the solution the information that obtains the zero-bit state parameter according to the coordinate information that obtains in the step 2.Concrete grammar is as follows:

Step 3.1, establish step 2.1 and be to the transformation matrix of step 2.2

Can be listed as by the equation that obtains in the step 1 (1.3) and to write following three equations:

(X ₁+A)cosθ ₁-Y ₁sinθ ₁+Q ₁sinθ ₁-P ₁cosθ ₁＝0 (2.1)

X ₂sinθ ₁+(Y ₂+A+B)cosθ ₁-P ₁sinθ ₁-Q ₁cosθ ₁＝0 (2.2)

X ₃sinθ ₁+(Y ₃+A)cosθ ₁-P ₁sinθ ₁-Q ₁cosθ ₁＝0 (2.3)

θ wherein ₁Be the angle that the moving coordinate system that forms in the step 2.2 is rotated counterclockwise with respect to the moving coordinate system in the step 2.1, P ₁Be the distance that the moving coordinate system that forms in the step 2.2 moves at the x direction of principal axis with respect to the moving coordinate system in the step 2.1, Q ₁The distance that moves at the y direction of principal axis with respect to the moving coordinate system in the step 2.1 for the moving coordinate system that forms in the step 2.2.

Step 3.2, known drive push-in stroke are Y under the zero zero-bit state ₂=Y ₃, the moving axes of establishing the machine initial point for (mXp mYp), then has:

$\left\{\begin{array}{c}\mathrm{mXp}=m{X}_1-X_1-A\\ \mathrm{mYp}=m{Y}_1-Y_2-A\end{array}\right.$

In the moving coordinate system of setting up because of step 1, the some moving axes before and after motion on the movable platform is constant, obtains the transformation relation of following zero-bit state and step 2.1 state coordinate system:

$\left[\begin{array}{c}m{X}_1-X_1-A\\ m{Y}_1-Y_1-\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">A</figure-callout>}\\ 1\end{array}\right]=\left[\begin{array}{ccc}\cos {\mathrm{\&theta;}}_1& -\sin {\mathrm{\&theta;}}_1& {\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1\sin {\mathrm{\&theta;}}_1-{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">P</figure-callout>}}_1\cos {\mathrm{\&theta;}}_1\\ \sin {\mathrm{\&theta;}}_1& \cos {\mathrm{\&theta;}}_1& -{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">P</figure-callout>}}_1\sin {\mathrm{\&theta;}}_1-{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1\cos {\mathrm{\&theta;}}_1\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}m{X}_2\\ m{Y}_2\\ 1\end{array}\right]$

Can obtain following two equations thus:

mX ₁-X ₁-A＝mX ₂cosθ ₁-mY ₂sinθ ₁+Q ₁sinθ ₁-P ₁cosθ ₁ (2.4)

mY ₁-Y ₁-A＝mX ₂sinθ ₁+mY ₂cosθ ₁-P ₁sinθ ₁-Q ₁cosθ ₁ (2.5)

Step 3.3, be always zero coordinate transform relation, establish by step 2.1 post exercise state and be to the transformation matrix of step 2.4 post exercise state according to the component of three change in coordinate axis direction of revolute pair center in the moving coordinate system of being set up

Then, can obtain following three equations as method described in 1:

(X ₁+A+B)cosθ ₂-Y ₁sinθ ₂+Q ₂sinθ ₂-P ₂cosθ ₂＝0 (2.6)

X ₂sinθ ₂+(Y ₂+A+2B)cosθ ₂-P ₂sinθ ₂-Q ₂cosθ ₂＝0 (2.7)

X ₃sinθ ₂+(Y ₃+A)cosθ ₂-P ₂sinθ ₂-Q ₂cosθ ₂＝0 (2.8)

Wherein, θ ₂Be the angle that the moving coordinate system that forms in the step 2.4 is rotated counterclockwise with respect to the moving coordinate system that forms in the step 2.1, P ₂The distance that the moving coordinate system that the moving coordinate system that forms for step 2.4 forms with respect to step 2.1 moves at the x direction of principal axis, Q ₂The distance that the moving coordinate system that the moving coordinate system that forms for step 2.4 forms with respect to step 2.1 moves at the y direction of principal axis.

Step 3.4, be similar to the method in above-mentioned 2, by the transformation relation of zero-bit state and step 2.4 state coordinate system, can be by transformation matrix T ₁Row are write two equations:

mX ₁-X ₁-A＝mX ₄cosθ ₂-mY ₄sinθ ₂+Q ₂sinθ ₂-P ₂cosθ ₂ (2.9)

mY ₁-Y ₁-A＝mX ₄sinθ ₂+mY ₄cosθ ₂-P ₂sinθ ₂-Q ₂cosθ ₂ (2.10)

Step 3.5, as can be known by Fig. 5 c, when step 2.3, do not change the corner of movable platform, and the linear drives push-in stroke of the active slip pair of second active branched chain 2# is 2B after step 2.4 is finished, therefore the corner of movable platform after step 2.4 is carried out is the twice that step 2.2 is carried out the back corner, that is:

θ ₂＝2θ ₁ (2.11)

And according to the configuration of 3-PRP planar three freedom parallel institution, knownly be under zero the zero-bit state driving push-in stroke,

Y ₂＝Y ₃ (2.12)

After obtaining (2.1)-(2.12) system of equations, can the solving equation group obtain (X ₁, Y ₁), (X ₂, Y ₂) and (X ₃, Y ₃), these expression formulas be input as four groups of machine origins and active drive amount A and the B that debugging step is measured gained:

${\mathrm{\&theta;}}_1=\arctan \frac{{\mathrm{mY}}_2-{\mathrm{mY}}_3}{{\mathrm{mX}}_3-{\mathrm{mX}}_2}$

${\mathrm{\&theta;}}_2=2\arctan \frac{{\mathrm{mY}}_2-{\mathrm{mY}}_3}{{\mathrm{mX}}_3-{\mathrm{mX}}_2}$

${\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">X</figure-callout>}}_1=\frac{({\mathrm{mY}}_4-{\mathrm{mY}}_2)\sin {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2+(A+B-{\mathrm{mX}}_4)\sin {\mathrm{\&theta;}}_1\cos {\mathrm{\&theta;}}_2+({\mathrm{mX}}_2-A)\cos {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2-({\mathrm{mX}}_1-A)(\sin {\mathrm{\&theta;}}_2-\sin {\mathrm{\&theta;}}_1)}{\sin ({\mathrm{\&theta;}}_2-{\mathrm{\&theta;}}_1)-({\sin \mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}$

${\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1=\frac{({\mathrm{mX}}_2-X_1-A)\cos {\mathrm{\&theta;}}_1+({\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">X</figure-callout>}}_1+A+B-{\mathrm{mX}}_4)\cos {\mathrm{\&theta;}}_2-{\mathrm{mY}}_2\sin {\mathrm{\&theta;}}_1+{\mathrm{mY}}_4\sin {\mathrm{\&theta;}}_2}{\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1}$

$X_2=\frac{(Y_2+A+B+{\mathrm{mY}}_2)\cos {\mathrm{\&theta;}}_1+(-Y_2-A-2B+{\mathrm{mY}}_4)\cos {\mathrm{\&theta;}}_2-{\mathrm{mX}}_2\sin {\mathrm{\&theta;}}_1+{\mathrm{mX}}_4\sin {\mathrm{\&theta;}}_2}{\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1}$

$Y_2=\frac{({\mathrm{mX}}_2-{\mathrm{mX}}_4)\sin {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2+(A+2B-{\mathrm{mY}}_4){\sin \mathrm{\&theta;}}_1\cos {\mathrm{\&theta;}}_2+({\mathrm{mY}}_2-A-B)\cos {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2-({\mathrm{mY}}_1-A)({\sin \mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}{\sin ({\mathrm{\&theta;}}_2-{\mathrm{\&theta;}}_1)-(\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}$

Y ₃＝Y ₂

$X_3=X_2+\frac{B}{\tan {\mathrm{\&theta;}}_1}$

The above-mentioned parameter that obtains solves the required zero-bit status information of orientation problem for the anti-method of separating.

Step 4, demarcation original state.

The anti-method of separating is in order to solve the 3-PRP parallel institution in actual use, how to obtain to arrive the required active drive amount of desired locations, realizing the Accurate Position Control to movable platform.In application of practical project, carry out that at every turn an original state is all arranged when the position is counter separates, this state is the state after motion that be set by the user or last exactly stops, thereby the push-in stroke that each active branched chain drives is also non-vanishing.So, counter the separating in the realization in the position, the demarcation of necessary advanced line home position.

Obtain in the step 3 being under the zero zero-bit state at the linear drives push-in stroke of three active branched chain, three revolute pair centers are with respect to the target coordinate (X that sits quietly ₁, Y ₁), (X ₂, Y ₂) and (X ₃, Y ₃).Three active branched chain that are set by the user during original state are a, b, c with respect to the linear advancement amount under the zero-bit state, and as shown in Figure 6, the mark of sitting quietly when the zero-bit state is Xp ₀Op ₀Yp ₀, the moving coordinate system during original state is Xp ₁Op ₁Yp ₁, be T to the initial transformation matrix of original state by the zero-bit state ₀,

${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HSA00000290529300031.png"\; state="\{\{state\}\}">T</figure-callout>}}_0=\left[\begin{array}{ccc}\cos \mathrm{\&theta;}& -\sin \mathrm{\&theta;}& Q\sin \mathrm{\&theta;}-P\cos \mathrm{\&theta;}\\ \sin \mathrm{\&theta;}& \cos \mathrm{\&theta;}& -P\sin \mathrm{\&theta;}-Q\cos \mathrm{\&theta;}\\ 0& 0& 1\end{array}\right]---\left(3.1\right)$

Wherein, θ is the angle that original state counterclockwise moves during with respect to the zero-bit state, the distance that P original state for respect to the zero-bit state time moves at the x direction of principal axis, the distance that Q original state for respect to the zero-bit state time moves at the y direction of principal axis.Three parameter values that can be obtained T0 by the equation of being set up in the step 1 (1.3) are:

$\mathrm{\&theta;}=\arctan \frac{Y_2-Y_3+b-c}{X_3-X_2}$

P＝(Y ₂+b-Y ₁)sinθcosθ+X ₂sin ²θ+(X ₁+a)cos ²θ)

Q＝-(X ₁+a-X ₂)sinθcosθ+Y ₁sin ²θ+(Y ₂+b)cos ²θ

Step 5, the position is counter separates.

The position is counter separates promptly according to the sit quietly mark (now value and expectation value) of two location points before and after motion of movable platform, obtains three linear drives amounts.

In order to realize that the 3-PRP parallel institution accurately locatees, need a kind of evaluation method and weigh whether accurate positioning of mechanism, the simplest method commonly used is judge mark point and whether arrives assigned address.And the 3-PRP parallel institution has three degree of freedom in plane motion, and the coordinate information that provides comprises translation and angle, and therefore, the calculation level that two marks are set can be forgiven these information.

As shown in Figure 6, the motion of parallel institution makes from original state moving coordinate system Xp ₁Op ₁Yp ₁Move to final position coordinate system Xp ₂Op ₂Yp ₂, the M point on the movable platform moves to the M2 place from M1, and the N point moves to the N2 place from N1.Obtaining initial transformation matrix T ₀The basis on, the anti-method of separating will be found the solution post exercise moving axes transformation matrix.Movable platform post exercise moving coordinate system is Xp ₂Op ₂Yp ₂, it is about the quiet coordinate system Xp of zero-bit ₀Op ₀Yp ₀Transformation matrix be T ₁,

${\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">T</figure-callout>}}_1=\left[\begin{array}{ccc}\cos (\mathrm{\&theta;}+r)& -\sin (\mathrm{\&theta;}+r)& S\sin (\mathrm{\&theta;}+r)-R\cos (\mathrm{\&theta;}+r)\\ \sin (\mathrm{\&theta;}+r)& \cos (\mathrm{\&theta;}+r)& -R\sin (\mathrm{\&theta;}+r)-S\cos (\mathrm{\&theta;}+r)\\ 0& 0& 1\end{array}\right]---\left(3.2\right)$

Wherein, r is the angle that counterclockwise moves when moving coordinate system is with respect to the zero-bit state after the motion, R is the distance that moves at the x direction of principal axis when moving coordinate system is with respect to the zero-bit state after the motion, and S is the distance that moves at the y direction of principal axis when moving coordinate system is with respect to the zero-bit state after the motion.

In the moving coordinate system of setting up because of step 1, the some moving axes before and after motion on the movable platform is constant, in conjunction with the available relation equation of equation (1.2) (3.3),

${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HSA00000290529300031.png"\; state="\{\{state\}\}">T</figure-callout>}}_0\left[\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="Mx"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Mx</figure-callout>}}_1\\ {\mathrm{My}}_1\\ 1\end{array}\right]={\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">T</figure-callout>}}_1\left[\begin{array}{c}{\mathrm{Mx}}_2\\ {\mathrm{My}}_2\\ 1\end{array}\right]---\left(3.3\right)$

Wherein, (Mx ₁, My ₁) and (Mx ₂, My ₂) be respectively the mark of sitting quietly before and after the motion of M point.

Be not difficult to try to achieve by the geometric relationship that Fig. 6 shows

Wherein, (Nx ₁, Ny ₁) and (Nx ₂, Ny ₂) be respectively the mark of sitting quietly before and after the motion of N point, with r substitution equation (3.3), solving equation (3.3) can obtain T again ₁Parameter:

R＝Gcos(θ+r)+Hsin(θ+r)，

S＝-Gsin(θ+r)+Hcos(θ+r)，

Wherein, G=-Mx ₁Cos θ+My ₁Sin θ-Qsin θ+Pcos θ+Mx ₂Cos (θ+r)-My ₂Sin (θ+r),

H＝Mx ₁sinθ-My ₁cosθ+Psinθ+Qcosθ+Mx ₂sin(θ+r)+My ₂cos(θ+r)，

In the above-mentioned expression formula, θ is the angle that the original state that solves of step 4 counterclockwise moves during with respect to the zero-bit state, the distance that moves at x axle, y direction of principal axis when P, Q represent original state with respect to the zero-bit state respectively.

If three linear drives advance Δ X respectively ₁, Δ Y ₂, Δ Y ₃, M1N1 is overlapped with M2N2, realize the motion of expectation.Then the mark of sitting quietly at back three the revolute pair centers of motion is respectively: (X ₁+ a+ Δ X ₁, Y ₁), (X ₂, Y ₂+ b+ Δ Y ₂) and (X ₃, Y ₃+ c+ Δ Y ₃).

At motion final position coordinate system Xp ₁Op ₁Yp ₁In, three revolute pair centers are still on the moving axes axle, and therefore same equation (1.3) can obtain resolving equation (3.4):

$\left\{\begin{array}{c}({\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">X</figure-callout>}}_1+a+{\mathrm{\&Delta;X}}_1)\cos (\mathrm{\&theta;}+r)-{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1\sin (\mathrm{\&theta;}+r)+S\sin (\mathrm{\&theta;}+r)-R\cos (\mathrm{\&theta;}+r)=0\\ X_2\sin (\mathrm{\&theta;}+r)+(Y_2+b+\mathrm{\&Delta;}{Y}_2)\cos (\mathrm{\&theta;}+r)-R\sin (\mathrm{\&theta;}+r)-R\cos (\mathrm{\&theta;}+r)=0\\ {X_3\sin (\mathrm{\&theta;}+r)+(Y_3+c+\mathrm{\&Delta;}{Y}_3)\cos (\mathrm{\&theta;}+r)-R\sin (\mathrm{\&theta;}+r)-S\cos }_{}(\mathrm{\&theta;}+r)=0\end{array}\right.---\left(3.4\right)$

So, try to achieve three active drive amounts smoothly, be anti-output of separating method and separate:

$\mathrm{\&Delta;}{\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">X</figure-callout>}}_1=\frac{{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="HSA00000290529300022.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1\sin (\mathrm{\&theta;}+r)-S\sin (\mathrm{\&theta;}+r)+R\cos (\mathrm{\&theta;}+r)}{\cos (\mathrm{\&theta;}+r)}-X_1-a,$

$\mathrm{\&Delta;}{Y}_2=\frac{-X_2\sin (\mathrm{\&theta;}+r)+R\sin (\mathrm{\&theta;}+r)+S\cos (\mathrm{\&theta;}+r)}{\cos (\mathrm{\&theta;}+r)}-Y_2-b,$

$\mathrm{\&Delta;}{Y}_3=\frac{-X_3\sin (\mathrm{\&theta;}+r)+R\sin (\mathrm{\&theta;}+r)+S\cos (\mathrm{\&theta;}+r)}{\cos (\mathrm{\&theta;}+r)}-Y_3-c.$

In actual applications, the anti-method of separating has successfully obtained realizing the required active drive amount Δ X of plane motion from the initial position (and nonzero digit position) that the user sets ₁, Δ Y ₂, Δ Y ₃Automatically drive amount is imported in the machine controller by control program, the controlling and driving motor is exported corresponding displacement again, can promote movable platform and realize accurately location.

Checking and embodiment

Adopt the method for virtual test,, come the correctness of localization method of the present invention is verified that verification step is as follows by setting up the motion of virtual prototype simulation mechanism:

(1) with the anti-program of separating method of C language compilation based on adjustment method, step 4 and the step 5 of above-mentioned steps two.

(2) in CAD/CAE software UG NX4, set up the model of 3-PRP planar three freedom parallel institution, as shown in Figure 2;

The coordinate of parallel institution model three revolute pair centers when the zero-bit state of (3) being set up among the logging software UG NX4 is used for the verifying and debugging method;

(4) in software UG NX4, mechanism model is carried out four debugging steps, obtain four groups of coordinates of the machine initial point of mark;

(5) push-in stroke of four groups of coordinates and debugging step is imported in the program of adjustment method, found the solution the zero-bit state parameter;

(6) compare the zero-bit state parameter and (3) middle write down whether consistent of trying to achieve in (5), then adjustment method is correct as if unanimity;

(7) in software UG NX4, mechanism model is provided with the push-in stroke of original state, obtains the coordinate of two gauge points original state before motion on the movable platform; The input motion drive amount obtains the coordinate of two gauge points after motion finishes again;

(8) push-in stroke of original state and two groups of coordinates of two gauge points will be set and import counter separating in the method program, find the solution three drive amount;

(9) whether the motion drive amount of the software input in the drive amount of relatively trying to achieve in (8) and (7) is consistent, correct as if the then anti-method of separating of unanimity.

Having provided a specific embodiment below verifies.

In the present embodiment, four groups of machine origins that obtain in (4) are respectively:

(-58499.8734778，-333058.2724765)，(-58690.6933958，-332564.6791878)，

(-58190.6942614，-332565.3370814)，(-58379.9011722，-332071.5906445)。

The zero-bit state parameter that solves in the program with their substitution adjustment methods, and the parameter of software records such as table 1 zero-bit state parameter record value and the value of the finding the solution table of comparisons in (3):

The table 1 zero-bit state parameter record value and the value of the finding the solution table of comparisons

Consider the numerical error of bringing because of the restriction of word length figure place when calculating, therefore the error of two groups of parameters is in tolerance interval, and adjustment method of the present invention can obtain the zero-bit state parameter of this parallel institution well.

The coordinate of two gauge points that obtain in the verification step (7) before and after motion is as shown in table 2:

The coordinate of record mark point before and after motion among the table 2 software UG NX4

The secondary motor1 of active slip of first active branched chain, the secondary motor2 of active slip of second active branched chain are respectively 2000mm, 5950mm and 3950mm with the initial propulsion amount of the secondary motor3 of active slip of the 3rd active branched chain.

The zero-bit state parameter that coordinate, initial propulsion amount and adjustment method program before and after the gauge point are solved is input in the anti-program of separating method, calculates three active drive amounts, and it is as shown in table 3 with the input drive amount contrast in (7):

The input motion drive amount among the table 3 software UG NX4 and the active drive amount table of comparisons that calculates

When calculating, the program of considering because the numerical error that the restriction of word length figure place brings, can think that therefore the error of two groups of parameters in tolerance interval, assert that the anti-method of separating can solve the orientation problem of this mechanism well.

Means by virtual test are carried out method validation, relatively it seems from result of calculation, and adjustment method and the anti-method of separating all are verified as correctly.Therefore, localization method of the present invention, the establishment, adjustment method and the anti-method of separating that comprise moving coordinate system and motion criterion, when solving the orientation problem of 3-PRP planar three freedom parallel institution, be effectively correct, can satisfy this mechanism working condition in actual applications fully, for the application of this mechanism provides feasible method.

Claims (4)

1. a 3-PRP planar three freedom parallel institution localization method is characterized in that, specifically may further comprise the steps:

Step 1, on the plane at place, three revolute pair centers, set up a plane moving coordinate system that together moves with motion platform with three revolute pair centers, the component of described three the some change in coordinate axis direction of revolute pair center in this moving coordinate system is always zero, the coordinate axis at the place, revolute pair center of first active branched chain is the y axle, and the coordinate axis at the place, revolute pair center of second and the 3rd active branched chain is the x axle;

Step 2, obtain machine initial point M by following debugging step ₀Under four kinds of states,, and write down these four marks of sitting quietly with respect to the mark of sitting quietly of quiet coordinate system; Described machine initial point M ₀It is the gauge point that is provided with in the center of rectangle movable platform;

Step 2.1, make the secondary all linear drives distance A of active slip of three active branched chain, obtain machine initial point M ₀Mark (the mX that sits quietly of this moment ₁, mY ₁);

Step 2.2, make second active branched chain the active slip pair again linear drives obtain machine initial point M apart from B ₀Sitting quietly of this moment is designated as (mX ₂, mY ₂);

Step 2.3, make first active branched chain the active slip pair again linear drives obtain machine initial point M apart from B ₀Sitting quietly of this moment is designated as (mX ₃, mY ₃);

Step 2.4, make second active branched chain the active slip pair again linear drives obtain machine initial point M apart from B ₀Sitting quietly of this moment is designated as (mX ₄, mY ₄);

Step 3, according to four that obtain in the step 2 marks of sitting quietly, according to the relation of coordinate system transformation, obtain the information of zero-bit state:

${\mathrm{\&theta;}}_1=\arctan \frac{{\mathrm{mY}}_2-{\mathrm{mY}}_3}{{\mathrm{mX}}_3-{\mathrm{mX}}_2}$

${\mathrm{\&theta;}}_2=2\arctan \frac{{\mathrm{mY}}_2-{\mathrm{mY}}_3}{{\mathrm{mX}}_3-{\mathrm{mX}}_2}$

$X_1=\frac{({\mathrm{mY}}_4-{\mathrm{mY}}_2)\sin {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2+(A+B-{\mathrm{mX}}_4)\sin {\mathrm{\&theta;}}_1\cos {\mathrm{\&theta;}}_2+({\mathrm{mX}}_2-A)\cos {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2-({\mathrm{mX}}_1-A)({\sin \mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}{\sin ({\mathrm{\&theta;}}_2-{\mathrm{\&theta;}}_1)-({\sin \mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}$

$Y_1=\frac{({\mathrm{mX}}_2-X_1-A)\cos {\mathrm{\&theta;}}_1+(X_1+A+B-{\mathrm{mX}}_4)\cos {\mathrm{\&theta;}}_2-{\mathrm{mY}}_2\sin {\mathrm{\&theta;}}_1+{\mathrm{mY}}_4\sin {\mathrm{\&theta;}}_2}{\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1}$

$X_2=\frac{(Y_2+A+B+{\mathrm{mY}}_2)\cos {\mathrm{\&theta;}}_1+(-Y_2-A-2B+{\mathrm{mY}}_4)\cos {\mathrm{\&theta;}}_2-{\mathrm{mX}}_2\sin {\mathrm{\&theta;}}_1+{\mathrm{mX}}_4\sin {\mathrm{\&theta;}}_2}{\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1}$

$Y_2=\frac{({\mathrm{mX}}_2-{\mathrm{mX}}_4)\sin {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2+(A+2B-{\mathrm{mY}}_4){\sin \mathrm{\&theta;}}_1\cos {\mathrm{\&theta;}}_2+({\mathrm{mY}}_2-A-B)\cos {\mathrm{\&theta;}}_1\sin {\mathrm{\&theta;}}_2-({\mathrm{mY}}_1-A)({\sin \mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}{\sin ({\mathrm{\&theta;}}_2-{\mathrm{\&theta;}}_1)-(\sin {\mathrm{\&theta;}}_2-{\sin \mathrm{\&theta;}}_1)}$

Y ₃＝Y ₂

$X_3=X_2+\frac{B}{\tan {\mathrm{\&theta;}}_1}$

Wherein, θ ₁The angle that the moving coordinate system that the moving coordinate system that forms for step 2.2 forms with respect to step 2.1 is rotated counterclockwise; θ ₂The angle that the moving coordinate system that the moving coordinate system that forms for step 2.4 forms with respect to step 2.1 is rotated counterclockwise; (X ₁, Y ₁), (X ₂, Y ₂) and (X ₃, Y ₃) be respectively three revolute pair centers coordinate with respect to quiet coordinate system when the zero-bit state;

The information of step 4, the zero-bit state that obtains according to step 3 is demarcated original state, and described original state is meant the state after motion that be set by the user or last stops; The original state that demarcation obtains is:

$\mathrm{\&theta;}=\arctan \frac{Y_2-Y_3+b-c}{X_3-X_2}$

P＝(Y ₂+b-Y ₁)sinθcosθ+X ₂sin ²θ+(X ₁+a)cos ²θ

Q＝-(X ₁+a-X ₂)sinθcosθ+Y ₁sin ²θ+(Y ₂+b)cos ²θ

Wherein, three active branched chain were with respect to the linear advancement amount under the zero-bit state when a, b, c were respectively the original state that the user sets, θ is the angle that original state counterclockwise moves during with respect to the zero-bit state, P is the distance that original state moves at the x direction of principal axis during with respect to the zero-bit state, and Q is the distance that original state moves at the y direction of principal axis during with respect to the zero-bit state;

Step 5, according to two location points mark of sitting quietly with respect to quiet coordinate system before and after movable platform motion, obtain with respect to three straight line active drive amounts under the original state:

$\mathrm{\&Delta;}{X}_1=\frac{Y_1\sin (\mathrm{\&theta;}+r)-S\sin (\mathrm{\&theta;}+r)+R\cos (\mathrm{\&theta;}+r)}{\cos (\mathrm{\&theta;}+r)}-X_1-a,$

$\mathrm{\&Delta;}{Y}_2=\frac{-X_2\sin (\mathrm{\&theta;}+r)+R\sin (\mathrm{\&theta;}+r)+S\cos (\mathrm{\&theta;}+r)}{\cos (\mathrm{\&theta;}+r)}-Y_2-b,$

$\mathrm{\&Delta;}{Y}_3=\frac{-X_3\sin (\mathrm{\&theta;}+r)+R\sin (\mathrm{\&theta;}+r)+S\cos (\mathrm{\&theta;}+r)}{\cos (\mathrm{\&theta;}+r)}-Y_3-c,$

Wherein, two location points are meant two the some M and the N of on movable platform mark, and the sitting quietly mark of some M before and after the movable platform motion is respectively (Mx ₁, My ₁) and (Mx ₂, My ₂), the sit quietly mark of some N before and after the movable platform motion is respectively (Nx ₁, Ny ₁) and (Nx ₂, Ny ₂), r is the angle that counterclockwise moves when moving coordinate system is with respect to the zero-bit state after the motion,

R is the distance that moves at the x direction of principal axis when moving coordinate system is with respect to the zero-bit state after the motion, and R=Gcos (θ+r)+Hsin (θ+r); S is the distance that moves at the y direction of principal axis when moving coordinate system is with respect to the zero-bit state after the motion, and S=-Gsin (θ+r)+Hcos (θ+r); Wherein, parameter G, H are respectively:

G＝-Mx ₁cosθ+My ₁sinθ-Qsinθ+Pcosθ+Mx ₂cos(θ+r)-My ₂sin(θ+r)，

H＝Mx ₁sinθ-My ₁cosθ+Psinθ+Qcosθ+Mx ₂sin(θ+r)+My ₂cos(θ+r)，

According to above-mentioned three straight line active drive amounts that obtain, the controlling and driving motor is exported corresponding displacement, promotes movable platform and realizes accurately location.

2. 3-PRP planar three freedom parallel institution localization method according to claim 1 is characterized in that, three revolute pair centers described in the step 1, the coordinate in the moving coordinate system of being set up be respectively (0, Yp ₁), (Xp ₂, 0) and (Xp ₃, 0), the relation according to coordinate system transformation obtains:

$\left\{\begin{array}{c}{\mathrm{Xp}}_1=(X_1+{\mathrm{\&Delta;X}}_1)\cos \mathrm{\&theta;}-Y_1\sin \mathrm{\&theta;}+Q^{\&prime;}\sin \mathrm{\&theta;}-P^{\&prime;}\cos \mathrm{\&theta;}=0\\ {\mathrm{Yp}}_2=X_2\sin \mathrm{\&theta;}+(Y_2+{\mathrm{\&Delta;Y}}_2)\cos \mathrm{\&theta;}-P^{\&prime;}\sin \mathrm{\&theta;}-Q^{\&prime;}\cos \mathrm{\&theta;}=0\\ {\mathrm{Yp}}_3=X_3\sin \mathrm{\&theta;}+\left(Y_3{+\mathrm{\&Delta;Y}}_3\right)\cos \mathrm{\&theta;}-P^{\&prime;}\sin \mathrm{\&theta;}-Q^{\&prime;}\cos \mathrm{\&theta;}=0\end{array}\right.$

Wherein, (Xp ₁, Yp ₁), (Xp ₂, Yp ₂) and (Xp ₂, Yp ₂) be respectively three coordinates of revolute pair center in moving coordinate system; (X ₁, Y ₁), (X ₂, Y ₂) and (X ₃, Y ₃) be respectively three initial positions of revolute pair center in quiet coordinate system; Δ X ₁, Δ Y ₂With Δ Y ₃Be respectively the push-in stroke of three active branched chain linear drives, θ is the angle that the moving coordinate system set up is rotated counterclockwise with respect to quiet coordinate system, and P ', Q ' are respectively the distance that the moving coordinate system set up moves along directions X, Y direction with respect to quiet coordinate system.

3. 3-PRP planar three freedom parallel institution localization method according to claim 1 is characterized in that the information of zero-bit state in the step 3 specifically obtains according to following steps:

Step 3.1, be always zero coordinate transform relation, to step 2.2 post exercise state, obtain following three formulas by step 2.1 post exercise state according to the component of three change in coordinate axis direction of revolute pair center in the moving coordinate system of being set up:

(X ₁+A)cosθ ₁-Y ₁sinθ ₁+Q ₁sinθ ₁-P ₁cosθ ₁＝0 (2.1)

X ₂sinθ ₁+(Y ₂+A+B)cosθ ₁-P ₁sinθ ₁-Q ₁cosθ ₁＝0 (2.2)

X ₃sinθ ₁+(Y ₃+A)cosθ ₁-P ₁sinθ ₁-Q ₁cosθ ₁＝0 (2.3)

Wherein, θ ₁Be the angle that the moving coordinate system that forms in the step 2.2 is rotated counterclockwise with respect to the moving coordinate system that forms in the step 2.1, P ₁The distance that the moving coordinate system that the moving coordinate system that forms for step 2.2 forms with respect to step 2.1 moves at the x direction of principal axis, Q ₁The distance that the moving coordinate system that the moving coordinate system that forms for step 2.2 forms with respect to step 2.1 moves at the y direction of principal axis;

Step 3.2, known drive push-in stroke are Y under the zero zero-bit state ₂=Y ₃, the moving axes of establishing the machine initial point for (mXp mYp), then has:

$\left\{\begin{array}{c}\mathrm{mXp}={\mathrm{mX}}_1-X_1-A\\ \mathrm{mYp}={\mathrm{mY}}_1-Y_2-A\end{array}\right.$

According to the some moving axes before and after motion on the movable platform is constant principle, obtains the transformation relation of following zero-bit state and step 2.1 state coordinate system:

$\left[\begin{array}{c}{\mathrm{mX}}_1-X_1-A\\ {\mathrm{mY}}_1-Y_1-A\\ 1\end{array}\right]=\left[\begin{array}{ccc}\cos {\mathrm{\&theta;}}_1& -\sin {\mathrm{\&theta;}}_1& Q_1\sin {\mathrm{\&theta;}}_1-P_1\cos {\mathrm{\&theta;}}_1\\ \sin {\mathrm{\&theta;}}_1& \cos {\mathrm{\&theta;}}_1& -P_1\sin {\mathrm{\&theta;}}_1-Q_1\cos {\mathrm{\&theta;}}_1\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathrm{mX}}_2\\ m{Y}_2\\ 1\end{array}\right]$

Obtain following two equations thus:

mX ₁-X ₁-A＝mX ₂cosθ ₁-mY ₂sinθ ₁+Q ₁sinθ ₁-P ₁cosθ ₁ (2.4)

mY ₁-Y ₁-A＝mX ₂sinθ ₁+mY ₂cosθ ₁-P ₁sinθ ₁-Q ₁cosθ ₁ (2.5)

Step 3.3, be always zero coordinate transform relation, to step 2.4 post exercise state, obtain following three formulas by step 2.1 post exercise state according to the component of three change in coordinate axis direction of revolute pair center in the moving coordinate system of being set up:

(X ₁+A+B)cosθ ₂-Y ₁sinθ ₂+Q ₂sinθ ₂-P ₂cosθ ₂＝0 (2.6)

X ₂sinθ ₂+(Y ₂+A+2B)cosθ ₂-P ₂sinθ ₂-Q ₂cosθ ₂＝0 (2.7)

X ₃sinθ ₂+(Y ₃+A)cosθ ₂-P ₂sinθ ₂-Q ₂cosθ ₂＝0 (2.8)

Wherein, θ ₂The angle that the moving coordinate system that the moving coordinate system that forms for step 2.4 forms with respect to step 2.1 is rotated counterclockwise, P ₂The distance that the moving coordinate system that the moving coordinate system that forms for step 2.4 forms with respect to step 2.1 moves at the x direction of principal axis, Q ₂The distance that the moving coordinate system that the moving coordinate system that forms for step 2.4 forms with respect to step 2.1 moves at the y direction of principal axis;

Step 3.4, with the method in the above-mentioned step 3.2, the transformation relation of the moving coordinate system that is formed by zero-bit state and step 2.4 obtains following two equations:

mX ₁-X ₁-A＝mX ₄cosθ ₂-mY ₄sinθ ₂+Q ₂sinθ ₂-P ₂cosθ ₂ (2.9)

mY ₁-Y ₁-A＝mX ₄sinθ ₂+mY ₄cosθ ₂-P ₂sinθ ₂-Q ₂cosθ ₂ (2.10)

Step 3.5, by geometric relationship, obtain the angle relation of moving coordinate system that step 2.4 forms and the moving axes of step 2.2 formation:

θ ₂＝2θ ₁ (2.11)

And according to the configuration of 3-PRP planar three freedom parallel institution, knownly be under zero the zero-bit state driving push-in stroke, have:

Y ₂＝Y ₃ (2.12)

The above-mentioned system of equations of simultaneous (2.1)-(2.12) obtain (X ₁, Y ₁), (X ₂, Y ₂) and (X ₃, Y ₃).

4. 3-PRP planar three freedom parallel institution localization method according to claim 1 is characterized in that, three straight line active drive amounts described in the step 5 specifically are to obtain by following process:

According to coordinate transform relation, the transformation matrix T of the quiet coordinate system when obtaining the post exercise moving coordinate system with respect to the zero-bit state ₁,

$T_1=\left[\begin{array}{ccc}\cos (\mathrm{\&theta;}+r)& -\sin (\mathrm{\&theta;}+r)& S\sin (\mathrm{\&theta;}+r)-R\cos (\mathrm{\&theta;}+r)\\ \sin (\mathrm{\&theta;}+r)& \cos (\mathrm{\&theta;}+r)& -R\sin (\mathrm{\&theta;}+r)-S\cos (\mathrm{\&theta;}+r)\\ 0& 0& 1\end{array}\right];$

According to the some moving axes before and after motion on the movable platform is constant principle, further obtains:

$T_0\left[\begin{array}{c}{\mathrm{Mx}}_1\\ {\mathrm{My}}_1\\ 1\end{array}\right]=T_1\left[\begin{array}{c}{\mathrm{Mx}}_2\\ {\mathrm{My}}_2\\ 1\end{array}\right]---(3.1)$

Wherein, T ₀Be the initial transformation matrix of zero-bit state to original state,

According to geometric relationship, obtain r, r substitution equation (3.1) is found the solution obtain T ₁Parameter G, H;

The sitting quietly of back three revolute pair centers of moving marked and is respectively: (X ₁+ a+ Δ X ₁, Y ₁), (X ₂, Y ₂+ b+ Δ Y ₂) and (X ₃, Y ₃+ c+ Δ Y ₃), be always zero coordinate transform relation according to the component of three change in coordinate axis direction of revolute pair center in the moving coordinate system of being set up, obtain:

$\left\{\begin{array}{c}(X_1+a+{\mathrm{\&Delta;X}}_1)\cos (\mathrm{\&theta;}+r)-Y_1\sin (\mathrm{\&theta;}+r)+S\sin (\mathrm{\&theta;}+r)-R\cos (\mathrm{\&theta;}+r)=0\\ X_2\sin (\mathrm{\&theta;}+r)+(Y_2+b+\mathrm{\&Delta;}{Y}_2)\cos (\mathrm{\&theta;}+r)-R\sin (\mathrm{\&theta;}+r)-R\cos (\mathrm{\&theta;}+r)=0\\ {X_3\sin (\mathrm{\&theta;}+r)+(Y_3+c+\mathrm{\&Delta;}{Y}_3)\cos (\mathrm{\&theta;}+r)-R\sin (\mathrm{\&theta;}+r)-S\cos }_{}(\mathrm{\&theta;}+r)=0\end{array}\right.---(3.2)$

Solving equation (3.2) obtains three active drive amount Δ X ₁, Δ Y ₂With Δ Y ₃

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