CN115922370A - Five-axis series-parallel motion machine tool and control method thereof - Google Patents
Five-axis series-parallel motion machine tool and control method thereof Download PDFInfo
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Abstract
The invention discloses a five-axis series-parallel motion machine tool and a control method thereof, belonging to the technical field of machine tools and particularly comprising a planar three-degree-of-freedom parallel mechanism, a bracket, a Y-direction guide rail assembly, a C-axis rotary table, a portal frame and a base. The invention can solve the problem of larger weight of the cantilever, can also realize vertical and horizontal processing conversion, has good integral rigidity, quick and sensitive processing and easy realization of high-precision processing, has good processing stability of the whole machine tool, relatively balanced stress of the inverted U-shaped bracket and higher precision when a Y-shaped guide rail assembly travels.
Description
Technical Field
The invention relates to a five-axis series-parallel motion machine tool and a control method thereof, belonging to the technical field of five-axis series-parallel motion machine tools.
Background
The parallel mechanism with less degrees of freedom is adopted as the main shaft head mechanism in the series-parallel motion machine tool, so that the advantages of the parallel motion mechanism can be exerted, and the parallel motion machine tool has the unique advantages of easiness in control, no motion coupling, large working space, simple structure, explicit expression of motion forward and inverse solution and the like. The main shaft head mechanism is an important component for developing a series-parallel motion machine tool, and generally comprises a feeding mechanism for driving a main shaft part to move and an electric main shaft. The general spindle head mechanism adopts a serial mechanism, the range of linear motion and rotary motion of the spindle head adopting the serial mechanism can be large, vertical and horizontal machining can be realized, but the cantilever of the spindle head mechanism has larger weight and high-speed feeding is limited.
In recent years, several complete parallel machine tools and parallel motion machine tools are successively introduced from country to country. The complete parallel machine tool has the outstanding advantages of small moving mass, high speed and high specific stiffness; the motion errors are not accumulated, and the precision is high. However, the operation range of the rotary motion is small, for example, a three-axis parallel mechanism is adopted by a Z3 main shaft head mechanism produced by DST company in Germany and Cincinnati company in America, the rotary range can reach +/-40 degrees at most, and the vertical and horizontal machining functions can not be realized.
The chinese patent application (application number 2013104526717) discloses a three-linear-drive planar three-degree-of-freedom parallel mechanism, which is not applicable to the parallel mechanism in the technical field of machine tools, and in a multi-axis hybrid motion machine tool, the motion is fast, high in rigidity and high in precision, and the fast, high in rigidity, high in precision and processing stability cannot be realized in the patent.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: provides a five-axis series-parallel motion machine tool, which aims to solve the technical problems in the prior art.
The technical scheme adopted by the invention is as follows: a five-axis series-parallel motion machine tool comprises a planar three-degree-of-freedom parallel mechanism, a bracket, a Y-direction guide rail assembly, a C-axis rotary table, a portal frame and a base, wherein the planar three-degree-of-freedom parallel mechanism is installed on the inner side of an inverted U-shaped bracket, the top of the inner side of the bracket is connected to the portal frame in a sliding mode through the Y-direction guide rail assembly and slides along the Y direction, the portal frame is fixedly connected to the base, the C-axis rotary table is installed on the base and located below the executing tail end of the planar three-degree-of-freedom parallel mechanism, and a power cutting head is installed at the executing end of the planar three-degree-of-freedom parallel mechanism.
Further, the planar three-degree-of-freedom parallel mechanism comprises a first linear moving assembly, a second linear moving assembly, a third linear moving assembly, a first connecting rod, a second connecting rod, a third connecting rod and a movable platform, wherein the first linear moving assembly is vertically arranged on the front side wall in the bracket, the second linear moving assembly and the third linear moving assembly are vertically arranged on the rear side wall in the bracket, the first connecting rod, the second connecting rod and the third connecting rod are equal in length, the upper ends of the first connecting rod, the second connecting rod and the third connecting rod are respectively hinged to sliders of the first linear moving assembly, the second linear moving assembly and the third linear moving assembly, the lower ends of the first connecting rod and the second connecting rod are respectively hinged to the front end of the movable platform, the lower end of the third connecting rod is hinged to the rear end of the movable platform, a mounting hole for mounting a power cutting head is formed in the middle of the movable platform, and the first linear moving assembly, the second linear moving assembly and the third linear moving assembly are respectively provided with a power device for driving the sliders to move.
Furthermore, the lower ends of the first connecting rod and the third connecting rod are respectively provided with an inverted U-shaped notch, the two sides of the U-shaped notch are hinged with the movable platform, the inner width of the U-shaped notch is larger than that of the movable platform, and the lower end of the second connecting rod is hinged at the left end of a hinged shaft between the first connecting rod and the movable platform.
Furthermore, the first connecting rod, the second connecting rod and the third connecting rod are hinged to the first linear moving assembly, the second linear moving assembly and the third linear moving assembly through hinge seats.
Furthermore, the portal frame comprises a top cross beam and two stand columns, the bottoms of the two ends of the top cross beam are respectively and fixedly connected to the upper ends of the two stand columns, and the lower ends of the two stand columns are fixedly connected to the base.
A control method of a five-axis series-parallel motion machine tool comprises the following steps: and simultaneously, the C-axis rotary table is controlled to rotate, the Y-direction guide rail component moving bracket and the planar three-degree-of-freedom parallel mechanism act to obtain the attitude of the moving platform, so that the attitude of the power cutting head is obtained, and the cutting processing of the workpiece is realized.
Five-axis series-parallel motion machineThe control method of the bed, the kinematic control method of the parallel mechanism of three degrees of freedom of the level is: three fixed-length connecting rods I, II and III are arranged as B i C i I =1,2,3, the sliding blocks of the two ends of the connecting rod connected with the movable platform and the fixed platform are respectively C 1 (C 2 )C 3 And A 1 A i A 3 Sliding block B i By linearly moving the assembly A i B i (i =1,2,3) is driven to move linearly, rod B i C i And a slide block B i Connected through a passive rotary hinge, the movable platform is arranged at C i The position is connected with each branch through a passive rotary joint to move the coordinate system sigma o of the platform m Built on a movable platform C 1 (C 2 )C 3 Geometric center position of (a), x m The shaft is vertical to the movable platform; z is a radical of m Shaft made of C 1 (or C 2 ) Point of direction C 3 The direction, the coordinate origin O of the fixed platform coordinate system sigma O is selected at the driving component A 1 B 1 Upper, X axis and A 1 B 1 The symmetric central lines of the two lines coincide, and the coordinate axes are positively arranged from A 1 Point of direction B 1 Z axis is defined by A 1 Point of direction A 3 ；
The kinematic position inverse solution method comprises the following steps: in the parallel mechanism, θ m X being the coordinate system of the moving platform m The rotation angle of the axis relative to the X axis of the fixed platform coordinate system, r is the radius of the movable platform, X 1 、X 2 And X 3 For input variable of the drive member, /) i Is B i C i (i =1,2,3) the length of the rods, then Σ O and Σ O m The homogeneous coordinate transformation matrix in between is:
in formula (1):
O R m is Σ o m With respect to the attitude matrix of Σ O, O P m is Σ o m Of the coordinate origin o m Coordinate representation in Σ O, where: c (theta) m )＝cos(θ m )，S(θ m )＝sin(θ m )；
C i Point (i =1,2,3) is at Σ o m The coordinates in (1) are respectively:
C 1 ：(0，0，-r)，C 2 ：(0，0，-r)，C 3 ：(0，0，r)
then C i The coordinates in Σ O of (i =1,2,3) are expressed as:
O C i ＝ O R m m C i + O P m (2)
to find C i The coordinates of the (i =1,2,3) point in Σ O are:
C 1 ：(x-rS(θ m ),0,z-rC(θ m ))
C 2 ：(x-rS(θ m ),0,z-rC(θ m ))
C 3 ：(x+rS(θ m ),0,z+rC(θ m ))
B i the coordinates of the (i =1,2,3) point in Σ O are:
B 1 ：(X 1 ,0,0) B 2 ：(X 2 ,0,a 1 ) B 2 ：(X 3 ,0,a 1 +a 2 )
A i the coordinates of the (i =1,2,3) point in Σ O are:
A 1 ：(0,0,0) A 2 ：(0,0,a 1 ) A 3 ：(0,0,a 1 +a 2 )
X 1 、X 2 and X 3 Is an input variable of the drive member, a 1 、a 2 Is the distance between the two guide rails.
The constraint equation of the parallel mechanism is as follows:
|B i -C i |＝l i (i＝1,2,3) (3)
thereby obtaining:
the inverse expression of the parallel mechanism can be obtained by the relation (4) as follows:
from the above relation, the inverse solution of the parallel mechanism has 8 groups of solutions, and in fact, due to the mechanism constraint, the motion of each active joint is on the left side of the movable platform, so that the formulas 6, 7 and 8 all take the "-" sign;
the kinematics position forward solution method comprises the following steps: c 1 (C 2 ) Coordinates of points (x) 1 ,0,z 1 ) From the input rod member A 1 B 1 、A 2 B 2 Input variable X of 1 、X 2 Determine, thereby x 1 、z 1 Are all variables X 1 、X 2 I.e.:
there is also the relation:
and theta m Is X 1 、X 2 、X 3 I.e.:
θ m ＝h(X 1 ,X 2 ,X 3 ) (10)
the kinematics of the mechanism is thus formulated as:
solving equation (11) to obtain:
in formula (13), a "+" sign is taken, wherein
N＝(X 1 -X 2 ) 2 +a 2
In equation (12), when X 1 ＝X 2 When the equation is solved, it is actually seen from the motion principle of the structure that in this case the mechanism is displaced in the X direction, so that a solution is actually present, and equation (11) is solved taking this into account:
in equation (15), since the movable platform is on the right side of the reference coordinate system, it takes a positive sign;
in the determination of x 1 、z 1 Then, solve for θ m The relation is obtained from equation (11):
k 1 S(θ m )+k 2 C(θ m )＝k 3 (16)
in the formula:
k 1 ＝x 1 -X 3
k 2 ＝z 1 -a 1 -a 2
thus, the following is obtained:
in formula (17), if k 1 ＝k 2 =0, the equation is not solved, in fact if k 1 =0, at this time rod B 3 C 3 The mechanism is vertical to the direction of the guide rail and generates a singular position, at the moment, the mechanism can not work normally, and the working state of avoiding the position can be set in motion control; if k is 2 =0, C of the mechanism at this time 1 (C 2 ) Point at guide rail A 3 B 3 Upper movement, which in fact does not occur in the practical application of the mechanism, the mechanism C 1 (C 2 ) The actual position of the point on the guide rail A 1 B 1 And a guide rail A 2 B 2 To (c) to (d);
in the formula (17), according to theta m Definition of the angle describing the rod C 1 C 3 An included angle between the normal direction and the positive direction of the X axis is determined through analysis, and a positive sign is taken at the included angle;
in the determination of x 1 、z 1 And theta m Then, the values of x and z are obtained by the equation (9);
the speed and acceleration analysis method of kinematics comprises the following steps: the time is differentiated on both sides of each equation in the system of equations (11) and is arranged as:
in the formula:
matrix J X The method comprises the following steps:
a 31 ＝x 1 +2rS(θ m )-X 3
a 32 ＝z 1 +2rC(θ m )-a 1 -a 2
a 33 ＝2r[C(θ m )(x 1 +2rS(θ m )-X 3 )-S(θ m )(z 1 +2rC(θ m )-a 1 -a 2 )]
if J X If | ≠ 0, then the Jacobian matrix of the parallel mechanism is obtained as:
whereby the speed of the parallel mechanism is positively solved as
If | J (q) | ≠ 0, the inverse of the speed of the parallel mechanism is found to be:
two derivatives are respectively obtained for time at two sides of each equation in the equation system (11) and are arranged as follows:
in the formula:
if J X If | ≠ 0, then the acceleration positive solution of the parallel mechanism is obtained as:
if J q | ≠ 0, solving the acceleration inverse solution of the parallel mechanism as follows:
the following equation (9) is obtained:
the solution method of the passive joint in the kinematics comprises the following steps: setting a slide block B at any time under the graphic coordinate system 1 、B 2 、B 3 Respectively move X 1 、X 2 、X 3 At this time, the movable platform coordinate system Sigma o m Has a coordinate origin in Σ O of (x, 0,z) sinceStructural parameter setting of parallel mechanism and B i 、C i The coordinate values of the (i =1,2,3) points have been given so that there are:
considering the right hand rule, then:
the invention has the beneficial effects that: compared with the prior art, the invention has the following effects:
(1) The five-axis series-parallel motion machine tool adopts a three-degree-of-freedom plane motion parallel mechanism as a main shaft head mechanism, can solve the problem of larger weight of the cantilever, can also realize vertical and horizontal processing conversion, has good integral rigidity, quick and sensitive processing and easy realization of high-precision processing, has good processing stability of the whole machine tool, is provided with an inverted U-shaped bracket and relatively balanced stress, and further has higher precision when a Y-shaped guide rail assembly moves;
(2) The three-dimensional numerical control machine tool is provided with three linear moving pairs P and 6 revolute pairs R (wherein C1 (C2) is a composite revolute pair formed by two revolute pairs), a cutting head on a movable platform realizes the processing of a complex curved surface and a complex part structure through the control of each linear moving assembly and each revolute assembly, the processing range is greatly increased, the three linear moving assemblies are connected with the movable platform to form three planar degrees of freedom, the overall support rigidity and stability are high, the response is sensitive and quick, the movable platform is supported by three connecting rods, and finally, force is transmitted to an inverted U-shaped bracket in the processing process, so that the stress of the U-shaped bracket is relatively balanced, the phenomenon that the connection stability between the bracket and a portal frame is poor due to the fact that the stress on one side is too large is avoided, the compensation of automatic stress can be realized, and the compensation of automatic motion errors is realized;
(3) The motion control method can lead the cutting head on the movable platform to realize the swinging and the walking in a large range, can realize the high-precision processing of the machine tool, has independent motions and generates interference in a complementary way, and has better motion stability.
Drawings
FIG. 1 is a schematic perspective view of the present invention;
FIG. 2 is a schematic perspective view of a planar three-degree-of-freedom parallel mechanism;
FIG. 3 is a schematic diagram of a parallel mechanism;
FIG. 4 is a schematic view of passive joint solving;
FIG. 5 is a schematic view of a workspace analysis;
FIG. 6 is a schematic diagram of the analysis of the rotation capability of the movable platform.
Detailed Description
The invention is further described with reference to the accompanying drawings and specific embodiments.
Example 1: as shown in fig. 1-2, a five-axis series-parallel motion machine tool includes a planar three-degree-of-freedom parallel mechanism 1, a bracket 2, a Y-direction guide rail assembly 3, a C-axis turntable 4, a gantry 5, and a base 6, where the planar three-degree-of-freedom parallel mechanism 1 is installed inside the inverted U-shaped bracket 2, the top of the inside of the bracket 2 is slidably connected to the gantry 5 through the Y-direction guide rail assembly 3 and slides along the Y direction, the gantry 5 is fixedly connected to the base 6, the C-axis turntable 4 is installed on the base 6 and is located below an execution end of the planar three-degree-of-freedom parallel mechanism 1, and an execution end of the planar three-degree-of-freedom parallel mechanism 1 is installed with a power cutting head.
The structure of the invention comprises three linear moving pairs P, 6 revolving pairs R (wherein C1 (C2) is a composite revolving pair consisting of two revolving pairs), a rod piece and a movable platform, wherein an electric spindle is arranged on the movable platform. The spindle head mechanism can realize linear motion in the direction of X, Z and B-axis motion rotating around the Y axis. The actuating end of the parallel mechanism is provided with a power cutting head, the planar three-degree-of-freedom parallel mechanism is arranged on the inner side of an inverted U-shaped bracket, the top of the inner side of the bracket is connected to a portal frame in a sliding mode through a Y-direction guide rail assembly and slides along the Y direction, the portal frame is fixedly connected to a base, and a C-axis rotary table is arranged on the base and located below the actuating tail end of the planar three-degree-of-freedom parallel mechanism.
The bracket 2 comprises a transverse sliding block part at the top and cantilever ends arranged at two ends of the sliding block part, the cantilever ends are of a gradual change structure from top to bottom, the components become small, the inner sides of the components are planes, the linear moving components can be conveniently arranged, and the structure ensures that the tumor is lighter under the condition of ensuring rigidity and stability;
further, the planar three-degree-of-freedom parallel mechanism 1 comprises a first linear moving assembly 101, a second linear moving assembly 102, a third linear moving assembly 103, a first connecting rod 104, a second connecting rod 105, a third connecting rod 106 and a movable platform 107, wherein the first linear moving assembly 101 is vertically installed on the front side wall in the bracket 2, the second linear moving assembly 102 and the third linear moving assembly 103 are vertically installed on the rear side wall in the bracket 2, the first connecting rod 104, the second connecting rod 105 and the third connecting rod 106 are equal in length, the upper ends of the first linear moving assembly 101, the second linear moving assembly 102 and the third linear moving assembly 103 are respectively hinged on sliding blocks, the lower ends of the first connecting rod 104 and the second connecting rod 105 are respectively hinged on the front end of the movable platform 107, the lower end of the third connecting rod 106 is hinged on the rear end of the movable platform 107, an installation hole 108 for installing a cutting head is formed in the middle of the movable platform 107, the first linear moving assembly 101, the second linear moving assembly 102 and the third linear moving assembly 103 are respectively provided with a power device for driving the sliding blocks to move, the parallel mechanisms are respectively and independently controlled, thereby realizing attitude control of the movable platform and attitude adjustment of power machining.
Furthermore, the lower ends of the first connecting rod 104 and the third connecting rod 106 are respectively provided with an inverted U-shaped notch 109, the movable platform 107 is hinged to two sides of the U-shaped notch 109, the width of the inner side of the U-shaped notch 109 is larger than that of the movable platform 107, the lower end of the second connecting rod 105 is hinged to the left end of a hinge shaft between the first connecting rod 104 and the movable platform 107, and the double-hinge mode is adopted, so that hinge stress is balanced, connection reliability and stability are better, and stable processing control is facilitated.
Furthermore, the first connecting rod 104, the second connecting rod 105 and the third connecting rod 106 are hinged to the first linear moving assembly 101, the second linear moving assembly 102 and the third linear moving assembly 103 through a hinge seat 110, the hinge seat 110 is fixedly connected to a sliding block of the linear moving assembly, each linear moving assembly comprises a guide rail, a sliding block, a screw-nut pair, a driving motor and a support, the two guide rails are adopted, the sliding blocks are slidably connected to the guide rails, the screw-nuts of the screw-nut pair are fixedly connected with the sliding blocks, two ends of each screw are rotatably connected to bearing seats, the bearing seats are fixedly connected to the support, the upper ends of the screws extend out and then are connected to the driving motor, and the driving motor is fixedly connected to the top end of the support through a motor frame.
Further, above-mentioned portal frame 5 includes crossbeam 501 and two stands 502 of top I shape, and crossbeam 501 is the base structure, and inside cavity, this structure light in weight, and the processing is consolidated in the place of inside expenditure guide rail, and overall rigidity is good, and fixed connection is respectively to two stands 502 upper ends bottom at top crossbeam 501 both ends, and two stands 502 lower extreme fixed connection are on base 6, and planer-type bearing structure, lathe rigidity and stability are better.
Example 2: a control method of a five-axis series-parallel motion machine tool comprises the following steps: and simultaneously, the C-axis rotary table is controlled to rotate, the Y-direction guide rail component moving bracket and the planar three-degree-of-freedom parallel mechanism 1 act to obtain the attitude of the moving platform, so that the attitude of the power cutting head is obtained, and the cutting processing of the workpiece is realized.
Mechanism description and degree of freedom calculation:
as shown in fig. 3, the parallel mechanism consists of three fixed length rods B i C i (i =1,2,3) connecting moving platform C 1 (C 2 )C 3 And a fixed platform A 1 A i A 3 Composition, slide block B i By linearly moving the assembly A i B i (i =1,2,3) is driven to move linearly, rod B i C i And a slide block B i Connected through a passive rotary hinge, the movable platform is arranged at C i Is connected with each branch through a passive rotary joint. For the convenience of solution and analysis, the coordinate system sigma o of the movable platform is processed m Is established on a figure moving platform C 1 (C 2 )C 3 Geometric center position of (1), x m The axis is vertical to the movable platform, and the direction is shown in the figure; z is a radical of m Shaft is composed of 1 (or C 2 ) Point of direction C 3 And (4) direction. Selecting the origin O of the sigma O coordinate system of the fixed platform on the driving component A 1 B 1 Upper, X axis and A 1 B 1 The symmetric central lines of the two lines coincide, and the coordinate axes are positively arranged from A 1 Point of direction B 1 Z axis is formed by 1 Point of direction A 3 。
As can be seen from fig. 3, the total number of components n =8 and the number of kinematic pairs g =9 in the parallel mechanism, and since the mechanism is a planar mechanism, there are three common constraints, i.e., λ =3, and thus the order d =6- λ =3, so that there are:
where M is the degree of freedom of the mechanism, f i The degree of freedom of the ith kinematic pair.
The mechanism is a plane three-freedom-degree parallel mechanism through calculation, and the degrees of freedom of the mechanism are linear movement along X and Z axes and rotation around Y axis. Since the parallel mechanism has P-R-R type branches, the mechanism is defined as a 3-PRR parallel mechanism, and the mechanism is subjected to kinematics-related problem solving.
The parallel mechanism kinematics analysis method comprises the steps of solving a position inverse solution, solving a position forward solution, analyzing speed and acceleration and solving a passive joint
(1) Solving for the inverse solution of position
As shown in FIG. 3, in the parallel mechanism, θ m X being a moving platform coordinate system m Relative axisAt the corner of the X axis of the fixed platform coordinate system, r is the radius of the movable platform, X 1 、X 2 And X 3 As input variable of the drive member, /) i Is B i C i (i =1,2,3) the length of the rods, then Σ O and Σ O m The homogeneous coordinate transformation matrix in between is:
in formula 2:
O R m is Σ o m With respect to the attitude matrix of sigma-O, O P m is Σ o m Of origin o m Coordinate representation in Σ O. In the formula: c (theta) m )＝cos(θ m )，S(θ m )＝sin(θ m )。
C i Point (i =1,2,3) is at Σ o m The coordinates in (1) are respectively:
C 1 ：(0，0，-r)C 2 ：(0，0，-r)C 3 ：(0，0，r)
then C is i The coordinates in Σ O of (i =1,2,3) are expressed as:
O C i ＝ O R m m C i + O P m (3)
can obtain C i The coordinates of the (i =1,2,3) point in Σ O are:
C 1 ：(x-rS(θ m ),0,z-rC(θ m ))
C 2 ：(x-rS(θ m ),0,z-rC(θ m ))
C 3 ：(x+rS(θ m ),0,z+rC(θ m ))
B i the coordinates of the (i =1,2,3) point in Σ O are:
B 1 ：(X 1 ,0,0) B 2 ：(X 2 ,0,a 1 ) B 2 ：(X 3 ,0,a 1 +a 2 )
A i the coordinates of the (i =1,2,3) point in Σ O are:
A 1 ：(0,0,0) A 2 ：(0,0,a 1 ) A 3 ：(0,0,a 1 +a 2 )
the constraint equation of the parallel mechanism is as follows:
|B i -C i |＝l i (i＝1,2,3) (4)
thereby, it is possible to obtain:
the inverse expression of the parallel mechanism can be obtained from relation 5 as follows:
from the above relation, it can be seen that the inverse solutions of the parallel mechanism have 8 groups of solutions, and in fact, due to the mechanism constraint, as shown in fig. 3, the motion of each active joint is on the left side of the moving platform, so that the formulas 6, 7 and 8 all take the "-" sign.
(2) Solving for positive solution of position
As can be seen in FIG. 3, C 1 (C 2 ) Coordinates of points (x) 1 ,0,z 1 ) From the input rod member A 1 B 1 、A 2 B 2 Input variable X of 1 、X 2 Determine, thereby x 1 、z 1 Are all variable X 1 、X 2 I.e.:
and has the relation:
and theta m Is X 1 、X 2 、X 3 I.e.:
θ m ＝h(X 1 ,X 2 ,X 3 ) (11)
the kinematic equation for the mechanism can thus be:
solving equation 12, it is easy to find:
in formula 14, a "+" sign is taken, wherein
N＝(X 1 -X 2 ) 2 +a 2
In equation 13, note that when X 1 ＝X 2 When the equation is solved, it is in fact clear from the principle of motion of the structure that, in this case, the mechanism is displaced in the X direction, so that in practice a solution exists, when consideredConsidering this factor, the following can be found by analyzing the scale 12:
in equation 16, since the movable platform is on the right side of the reference coordinate system, it takes a positive sign.
In the determination of x 1 、z 1 Then, the solution of θ can be obtained m From equation 12, the relationship:
k 1 S(θ m )+k 2 C(θ m )＝k 3 (17)
in formula 17:
k 1 ＝x 1 -X 3
k 2 ＝z 1 -a 1 -a 2
thus, it is possible to obtain:
in equation 18, note that if k 1 ＝k 2 =0, the equation is solved. In fact if k 1 =0, at this time rod B 3 C 3 The mechanism is vertical to the direction of the guide rail and generates a singular type position, and at the moment, the mechanism cannot work normally, and the working state of the position can be avoided in motion control. If k is 2 =0, C of the mechanism at this time 1 (C 2 ) Point at guide rail A 3 B 3 Upper movement, which in fact does not occur in the practical application of the mechanism, the mechanism C 1 (C 2 ) Actual position of occurrence of pointIs arranged on the guide rail A 1 B 1 And a guide rail A 2 B 2 In the meantime.
In the formula 18, according to theta m Definition of the angle describing the rod C 1 C 3 The included angle between the normal direction and the positive direction of the X axis can be determined through simulation analysis, and the included angle is a positive sign.
In the determination of x 1 ,z 1 ，θ m Then, the values of x and z can be obtained by equation 10.
(3) Velocity and acceleration analysis
The time is differentiated on both sides of each equation in the system of equations 12 and is arranged as:
in formula 19:
matrix J X The method comprises the following steps:
a 31 ＝x 1 +2rS(θ m )-X 3
a 32 ＝z 1 +2rC(θ m )-a 1 -a 2
a 33 ＝2r[C(θ m )(x 1 +2rS(θ m )-X 3 )-S(θ m )(z 1 +2rC(θ m )-a 1 -a 2 )]
if J X If | ≠ 0, then the Jacobian matrix of the parallel mechanism can be obtained as follows:
whereby the speed of the parallel mechanism is positively solved as
If | J (q) | ≠ 0, the inverse of the speed of the parallel mechanism can be found as:
the time is divided into two derivatives on both sides of each equation in the equation set 12, and the two derivatives are arranged as:
in formula 23:
if J X If | ≠ 0, then the acceleration positive solution of the parallel mechanism can be obtained as:
if J q If | ≠ 0, the inverse solution of the acceleration of the parallel mechanism can be obtained as:
it can be easily obtained from equation 10:
(4) Passive joint solving
As shown in FIG. 4, in the coordinate system shown in the figure, a slide block B is set at an arbitrary time 1 、B 2 、B 3 Respectively move X 1 、X 2 、X 3 At this time, the movable platform coordinate system Sigma o m Has the coordinate origin of (x, 0,z) in sigma O, and B is given by the structural parameters of the parallel mechanism i 、C i The coordinate values of the (i =1,2,3) points have been given above, so that there are:
considering the right hand rule, then:
the parallel mechanism performance analysis method comprises the following steps: comprises the steps of working space analysis, singularity analysis and moving platform rotation capacity analysis
(1) And (3) analyzing a working space: for convenience of explanation, here is denoted as C 1 (C 2 ) Points as reference points for the moving platform, according to the preceding analysis, C 1 (C 2 ) Point by equation (x) 1 -X 1 ) 2 +z 1 2 ＝l 1 2 And (x) 1 -X 2 ) 2 +(z 1 -a 1 ) 2 ＝l 2 2 Determined together, C is readily apparent 1 (C 2 ) Is characterized by (X) 1 ,0)、(X 2 ,a 1 ) Respectively as the center of circle with 1 1 、l 2 Respectively, the intersection of two circles of radius. If X is i (i =1,2) in interval [0,L]In a variation of C 1 (C 2 ) The set of points is (X) 1 ,0)、(X 2 ,a 1 ) Respectively as the center of a circle 1 、l 2 The two circles with the respective radii roll along the X direction to form an intersection of enveloping surfaces. As shown in FIG. 5, given any attitude angle θ m In the initial state, the positional relationship of the parallel mechanism is shown by the black solid line in the figure, and the slider B 3 Moved along the guide by a distance d, while the slide B 1 、B 2 At the initial position, as can be seen from equation 12, C is given the attitude of the moving platform 1 (C 2 ) The positional relationship of the points is also given by the equation (x) 1 +2rS(θ m )-X 3 ) 2 +(z 1 +2rC(θ m )-a 1 -a 2 ) 2 ＝l 3 2 The equation describes, in fact, one or more (X) 3 -2rS(θ m ),a 1 +a 2 -2rC(θ m ) Is centered at l) 3 Is a circle with a radius, and the posture of the movable platform is given, so that when the sliding block B is in a sliding state 3 In-line assembly A 3 B 3 Up movement X 3 While the Z coordinate value of the center of the circle remains unchanged, when the slide block B is used 3 In the interval [0,L]In the upper variation, the equation (x) 1 +2rS(θ m )-X 3 ) 2 +(z 1 +2rC(θ m )-a 1 -a 2 ) 2 ＝l 3 2 The determined circle is just on the guide rail A 4 B 4 (the rail does not actually exist and is an equation-dependent virtual rail, as shown by the dashed line in fig. 5) to form an envelope surface. Whereby C takes into account the attitude of the moving platform 1 (C 2 ) The set of points is three circles determined by equation set 12 along the rail in the interval [0,L ]]The intersection of the envelope surfaces formed in the upper variation, as indicated by the hatched portion in the figure.
Under the condition that the attitude of the movable platform is fixed, the value o can be solved through simple coordinate transformation m For the working space when the moving platform is referenced, it is obvious that the working space determined when the attitude of the moving platform is given is a subspace of the reachable working space. In the parallel mechanism discussed in this invention, when X is 1 、X 2 In certain cases by varying X 3 The change of the attitude of the movable platform can be realized.
(2) Singularity analysis
The singularity problem is an important characteristic of the parallel mechanism, when the parallel mechanism is in a singularity state, the mechanism motion is out of control or control failure is caused, and an expected target cannot be achieved.
(1) If J q I =0, and | J X And | ≠ 0. At the moment, the parallel mechanism is positioned at the boundary of the working space, namely the boundary of the working space is singular. The following relationship can be obtained by solving:
(x 1 -X 1 )(x 1 -X 2 )(x 1 +2rS(θ m )-X 3 )＝0 (34)
the above relation shows the rod B of the parallel mechanism i C i In the case where any one of the rods (i =1,2,3) is perpendicular to the X coordinate direction of Σ O, the mechanism is in a workspace boundary singular state.
(2) If J X I =0, and | J q And | ≠ 0. At the moment, the parallel mechanism is positioned in a working space, the degree of freedom of the mechanism is increased, the movable platform cannot bear any external load, namely the movable platform is singular in the working space, and the following relational expression can be obtained by solving:
(x 1 -X 1 )(z 1 -a 1 )-z 1 (x 1 -X 2 )＝0 (35)
or
C(θ m )(x 1 +2rS(θ m )-X 3 )-S(θ m )(z 1 +2rC(θ m )-a 1 -a 2 )＝0 (36)
The above relation shows the rod member B of the parallel mechanism 1 C 1 And B 2 C 2 In a straight line or bar B 3 C 3 And C 2 C 3 When on a straight line, the parallel mechanism is in a singular state in a working space. As shown in fig. 3, this singular condition does not actually occur due to mechanical structural constraints.
(3) If J X I =0, and | J q L =0. At this time, the parallel mechanism has a singular structure, and the following situations can be seen from formulas 34 to 36: 1) The rod member B of the parallel mechanism 1 C 1 And B 2 C 2 Both in a straight line and perpendicular to the X-axis, when there is a relation a 1 ＝l 1 +l 2 As can be seen from the analysis of the known art, this situation is to be avoided at the design stage. 2) Rod member B of the parallel mechanism 1 C 1 And B 2 C 2 Is perpendicular to the X-axis and the bar B 3 C 3 And C 2 C 3 On a straight line. 3) The rod member B of the parallel mechanism 3 C 3 And C 2 C 3 Both in a straight line and perpendicular to the X-axis, when B is present 3 C 3 Or C 2 C 3 The length of the parallel mechanism is zero, and the parallel mechanism with three degrees of freedom researched by the invention is changed into a parallel mechanism with two degrees of freedom of a plane and redundant drive. 4) Rod member B of the parallel mechanism 1 C 1 And B 2 C 2 In a straight line and a rod member B 3 C 3 Perpendicular to the X-axis. When the mechanism is different in structure, the movable platform can still have slight movement under the condition that the mechanism has no input, and the driving mechanism still has slight input under the condition that the movable platform has no output.
Analyzing the rotation capacity of the movable platform: as shown in FIG. 6, the schematic configuration of the parallel mechanism in the initial state is shown, where θ is the vertical direction of the movable platform of the parallel mechanism in the X coordinate direction of Σ O in the initial state m =0, and stipulates that B is present at this time i And A i Coincidence, i.e. X i ＝0(i＝1，2，3)。
The rotating capability of the movable platform is analyzed, so that the C is appointed 1 (C 2 ) The point remains unchanged so that C 3 Point winding C 1 (C 2 ) Point with B 3 The point rotates along the movement of the guide rail. When B is present 3 The point moves along the guide rail d 1 To B' 3 Point-time, moving platform C 1 (C 2 )C 3 Go to C 1 (C 2 )C′ 3 At position, at this time θ m =90 °; when B is present 3 The point continues to move along the guide rail d 2 Reach B ″ 3 Point-time, moving platform C 1 (C 2 )C 3 Go to C 1 (C 2 )C″ 3 In position, the movable platform is now rotated through an angle of alpha again, in which state the rod B 3 C 3 The direction of the coordinate is vertical to the X coordinate direction of the sigma-O, and the parallel mechanism is in a singular state; when B is present 3 The point continues moving along the guide rail d 3 To B' 3 Point-time, moving platform C 1 (C 2 )C 3 From C 1 (C 2 )C″ 3 Position rotation beta angle to C 1 (C 2 )C″′ 3 At position of the bar member B 3 C 3 And movable platform rod C 2 C 3 On a straight line, B 3 The point moves along the guide rail to the extreme position, and the parallel mechanism is in a singular state.
From the above analysis, it can be seen that the maximum angle that the parallel mechanism movable platform can rotate in the initial state is 90 ° + α, and generallyThe movable platform is expected to realize vertical and horizontal machining states, and in order to realize the conversion of vertical and horizontal machining, the parallel mechanism is in an initial state B 3 The point needs to move along the guide rail d 1 The following can be obtained easily:
in the formula 37, the symbol "-" is used.
It can be seen from the working space analysis that, due to the influence of the attitude of the moving platform, the space determined when the attitude of the moving platform is given is often a subspace of the reachable space of the parallel mechanism, and therefore, to study the rotation capability of the parallel mechanism, it is actually to determine the attitude space of the parallel mechanism, that is, the rotation capability in a certain attitude space. d 1 The determination of (b) is advantageous for selecting the travel of the guide rail.
Static analysis of the parallel mechanism: if with C 1 (C 2 ) The point is a movable platform reference point and is arranged at C of a movable platform of the parallel mechanism 1 (C 2 ) The effect of the applied external load F at the point is noted as: f = [ F = X ,F Z ,M Y ] T The driving force of each driving joint is τ, where τ = [ τ = [ [ τ ] 1 ,τ 2 ,τ 3 ] T . According to the virtual work principle, the sum of the virtual work done by each driving joint and the sum of the virtual work done by the movable platform should be equal, so that the following relation holds:
in the formula 38, the compound represented by the formula,
to correspond to the virtual displacement of the joint, delta, of the driving force X 、δ Z 、δ θ Is the virtual displacement corresponding to the external load of the movable platform. From the arbitrariness of the virtual displacement, the statics inverse solution of the parallel mechanism is easily found as:
τ＝J(q) T F (39)
J(q) T for the transpose matrix of the mechanism speed Jacobian matrix, the statics positive solution of the parallel mechanism is easily obtained by using the inverse matrix of the Jacobian matrix as follows:
F＝(J(q) T ) -1 τ＝(J(q) -1 ) T τ (40)
the driving force and driving moment required by the parallel mechanism under the condition of the known external load action of the movable platform can be obtained by utilizing an inverse solution solving formula, and the bearing condition of the movable platform of the parallel mechanism under the condition of the known driving force and driving moment can be obtained by utilizing a forward solution solving formula.
The above description is only an embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily think of the changes or substitutions within the technical scope of the present invention, and therefore the scope of the present invention shall be subject to the protection scope of the claims.
Claims (7)
1. A five-axis series-parallel motion machine tool is characterized in that: including three degree of freedom parallel mechanism in plane (1), bracket (2), Y leads rail set spare (3), C axle revolving platform (4), portal frame (5) and base (6), three degree of freedom parallel mechanism in plane (1) are installed at bracket (2) inboard of the U type of handstand, bracket (2) inboard top is passed through Y and is led rail set spare (3) sliding connection on portal frame (5) and slide along the Y to, portal frame (5) fixed connection is on base (6), C axle revolving platform (4) are installed on base (6) and are located the terminal below of execution of three degree of freedom parallel mechanism in plane (1), the execution end of three degree of freedom parallel mechanism in plane (1) is installed the power cutting head.
2. The five-axis series-parallel motion machine tool according to claim 1, characterized in that: the planar three-degree-of-freedom parallel mechanism (1) comprises a first linear moving assembly (101), a second linear moving assembly (102), a third linear moving assembly (103), a first connecting rod (104), a second connecting rod (105), a third connecting rod (106) and a movable platform (107), wherein the first linear moving assembly (101) is vertically installed on the inner front side wall of a bracket (2), the second linear moving assembly (102) and the third linear moving assembly (103) are vertically installed on the inner rear side wall of the bracket (2), the lengths of the first connecting rod (104), the second connecting rod (105) and the third connecting rod (106) are equal, the upper ends of the first connecting rod (104), the second connecting rod (105) and the third connecting rod (106) are respectively hinged on sliders of the first linear moving assembly (102), the second linear moving assembly (102) and the third linear moving assembly (103), the lower ends of the first connecting rod (104) and the second connecting rod (105) are respectively hinged at the front end of the movable platform (107), the lower end of the third connecting rod (106) is hinged at the rear end of the movable platform (107), a mounting hole (108) for mounting a power cutting head is formed in the middle of the movable platform (107), and power driving devices are respectively arranged on the sliders of the first linear moving assembly (101), the second linear moving assembly (102), the second linear moving assembly (103).
3. The five-axis series-parallel motion machine tool according to claim 2, characterized in that: the lower ends of the first connecting rod (104) and the third connecting rod (106) are respectively provided with an inverted U-shaped notch (109), two sides of the U-shaped notch (109) are hinged to the movable platform (107), the width of the inner side of the U-shaped notch (109) is larger than that of the movable platform (107), and the lower end of the second connecting rod (105) is hinged to the left end of a hinge shaft between the first connecting rod (104) and the movable platform (107).
4. The five-axis series-parallel motion machine tool according to claim 2, characterized in that: the first connecting rod (104), the second connecting rod (105) and the third connecting rod (106) are hinged to the first linear moving component (101), the second linear moving component (102) and the third linear moving component (103) through hinge seats (110).
5. The five-axis series-parallel motion machine tool according to claim 2 or 5, characterized in that: the portal frame (5) comprises a top cross beam (501) and two columns (502), the bottoms of the two ends of the top cross beam (501) are respectively and fixedly connected to the upper ends of the two columns (502), and the lower ends of the two columns (502) are fixedly connected to the base (6).
6. The control method of a five-axis series-parallel motion machine tool according to any one of claims 1 to 5, characterized in that: the method comprises the following steps: and simultaneously, the C-axis rotary table is controlled to rotate, the Y-direction guide rail component moving bracket and the planar three-degree-of-freedom parallel mechanism (1) are controlled to act, so that the attitude of the movable platform is obtained, the attitude of the power cutting head is further obtained, and the cutting processing of the workpiece is realized.
7. The control method of the five-axis series-parallel motion machine tool according to claim 6, characterized in that: the kinematic control method of the planar three-degree-of-freedom parallel mechanism (1) comprises the following steps: three fixed-length connecting rods I, II and III are arranged as B i C i I =1,2,3, the sliding blocks at the two ends of the connecting rod connected with the movable platform and the fixed platform are respectively C 1 (C 2 )C 3 And A 1 A 2 A 3 Sliding block B i By linearly moving the assembly A i B i (i =1,2,3) is driven to move linearly, rod B i C i And a slide block B i Connected through a passive rotary hinge, the movable platform is arranged at C i The position is connected with each branch through a passive rotary joint to move the coordinate system sigma o of the platform m Built on a movable platform C 1 (C 2 )C 3 Geometric center position of (1), x m The shaft is vertical to the movable platform; z is a radical of m Shaft is composed of 1 (or C 2 ) Point of direction C 3 The direction, the coordinate origin O of the fixed platform coordinate system sigma O is selected at the driving component A 1 B 1 Upper, X axis and A 1 B 1 The symmetric central lines of the two lines coincide, and the coordinate axes are positively arranged from A 1 Point of direction B 1 Z axis is defined by A 1 Point of direction A 3 ；
The kinematic position inverse solution method comprises the following steps: in the parallel mechanism, θ m X being a moving platform coordinate system m The rotation angle of the axis relative to the X axis of the coordinate system of the fixed platform, r is the radius of the movable platform, X 1 、X 2 And X 3 For input variable of the drive member, /) i Is B i C i (i =1,2,3) the length of the rods, then Σ O and Σ O m The homogeneous coordinate transformation matrix in between is:
in formula (1):
O R m is Σ o m With respect to the attitude matrix of sigma-O, O P m is Σ o m Of the coordinate origin o m Coordinate representation in Σ O, where: c (theta) m )＝cos(θ m )，S(θ m )＝sin(θ m )；
C i Point (i =1,2,3) is at Σ o m The coordinates in (1) are respectively:
C 1 ：(0，0，-r)，C 2 ：(0，0，-r)，C 3 ：(0，0，r)
then C i The coordinates of (i =1,2,3) in Σ O are expressed as:
O C i ＝ O R m m C i + O P m (2)
to find out C i The coordinates of the (i =1,2,3) point in Σ O are:
C 1 ：(x-rS(θ m ),0,z-rC(θ m ))
C 2 ：(x-rS(θ m ),0,z-rC(θ m ))
C 3 ：(x+rS(θ m ),0,z+rC(θ m ))
B i the coordinates of the (i =1,2,3) point in Σ O are:
B 1 ：(X 1 ,0,0)，B 2 ：(X 2 ,0,a 1 )，B 2 ：(X 3 ,0,a 1 +a 2 )
X 1 、X 2 and X 3 As input variable of the drive member, a 1 、a 2 Is the distance between the two guide rails;
A i the coordinates of the (i =1,2,3) point in Σ O are:
A 1 ：(0,0,0) A 2 ：(0,0,a 1 ) A 3 ：(0,0,a 1 +a 2 )
the constraint equation of the parallel mechanism is as follows:
|B i -C i |＝l i (i＝1,2,3) (3)
thereby obtaining:
the inverse expression of the parallel mechanism can be obtained by the relation (4) as follows:
from the above relation, the inverse solution of the parallel mechanism has 8 groups of solutions, and in fact, due to the mechanism constraint, the motion of each active joint is on the left side of the movable platform, so that the formulas (5), (6) and (7) all take a "-";
the kinematics position forward solution method comprises the following steps: c 1 (C 2 ) Coordinates of points (x) 1 ,0,z 1 ) From the input rod A 1 B 1 、A 2 B 2 Input variable X of 1 、X 2 Determine, thereby x 1 、z 1 Are all variable X 1 、X 2 I.e.:
and has the relation:
and theta m Is X 1 、X 2 、X 3 I.e.:
θ m ＝h(X 1 ,X 2 ,X 3 ) (10)
the kinematics of the mechanism is thus formulated as:
solving equation (11) to obtain:
in formula (13), the symbol is "+", wherein
N＝(X 1 -X 2 ) 2 +a 2
In equation (12), when X 1 ＝X 2 When the equation is solved, it is in fact seen by the principle of motion of the structure, in which case the mechanism is displaced in the X direction, so that in fact a solution exists, taking into account this factor the equation (11)) And (3) analysis and calculation:
in equation (15), since the movable platform is on the right side of the reference coordinate system, the positive sign is taken;
in the determination of x 1 、z 1 Then, solve for θ m The relation is obtained from equation (11):
k 1 S(θ m )+k 2 C(θ m )＝k 3 (16)
in the formula:
k 1 ＝x 1 -X 3
k 2 ＝z 1 -a 1 -a 2
thus, the following is obtained:
in formula (17), if k 1 ＝k 2 =0, the equation is solved without, in fact, if k 1 =0, at this time rod B 3 C 3 The mechanism generates a singular type position in a direction vertical to the guide rail; if k is 2 =0, C of the mechanism at this time 1 (C 2 ) Point at guide rail A 3 B 3 Upper movement, which in fact does not occur in the practical application of the mechanism, the mechanism C 1 (C 2 ) The actual position of the point on the guide rail A 1 B 1 And a guide rail A 2 B 2 To (c) to (d);
in the formula (17), according to theta m Definition of the angle describing the rod C 1 C 3 The included angle between the normal direction and the positive direction of the X axis is taken as a positive sign;
in the determination of x 1 、z 1 And theta m Then, the values of x and z are obtained by the equation (9);
the speed and acceleration analysis method of kinematics comprises the following steps: the time is differentiated on both sides of each equation in the system of equations (11) and is arranged as:
in the formula:
matrix J X The method comprises the following steps:
a 31 ＝x 1 +2rS(θ m )-X 3
a 32 ＝z 1 +2rC(θ m )-a 1 -a 2
a 33 ＝2r[C(θ m )(x 1 +2rS(θ m )-X 3 )-S(θ m )(z 1 +2rC(θ m )-a 1 -a 2 )]
if J X If | ≠ 0, then the Jacobian matrix of the parallel mechanism is obtained as:
whereby the speed of the parallel mechanism is positively solved as
If | J (q) | ≠ 0, the inverse of the speed of the parallel mechanism is found to be:
two derivatives are respectively obtained for time at two sides of each equation in the equation system (11) and are arranged as follows:
in the formula:
if J X If | ≠ 0, then the acceleration positive solution of the parallel mechanism is obtained as:
if J q | ≠ 0, solving the acceleration inverse solution of the parallel mechanism as follows:
the following equation (9) is obtained:
the solution method of the passive joint in the kinematics comprises the following steps: setting a slide block B at any time under the graphic coordinate system 1 、B 2 、B 3 Respectively move X 1 、X 2 、X 3 At this time, the movable platform coordinate system Sigma o m Has the coordinate origin of (x, 0,z) in sigma O, and B is given by the structural parameters of the parallel mechanism i 、C i The coordinate values of the (i =1,2,3) points have been given so that there are:
considering the right hand rule, then:
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