KR20050022756A - Parallel Typed Machining Tool for Complex 3D Machining Work - Google Patents

Abstract

PURPOSE: A parallel-type machining tool for machining composite shapes is provided to execute correction work by regulating freely the height of a constrained plane and examining observability of a constrained movement by one or three digital indicators and one constrained surface and to search an optimal correction shape for maximizing independence of a line vector by using QR-decomposition. CONSTITUTION: In a parallel-type machining tool for machining composite shapes, observability is examined according to constrained movement and an optimal correction shape is discovered. A non-observable variable is estimated and the coordinate system is found out. Plane constraining movement uses one or three digital indicators and one constrained plane. The optimal correction shape maximizes independence of a line vector by QR-decomposing an observation matrix. The number of observable variables is provided from a rank of the observation matrix. A value for indicating independence of the line vector is defined and minimized to examine observability according to a constraint variable.

Description

Parallel Type Machine Tool for Complex Machining {Parallel Typed Machining Tool for Complex 3D Machining Work}

The correction is to increase the absolute precision to the repeatability and to find and correct the factors that hinder the precision. In general, the calibration process interprets kinematics, finds the errors due to the difference between actual and theoretical motions, and finds the actual dimensions that minimize them. Among the above phases, the main issue is the measurement technology and the observability of kinematic variables that accurately measure actual motion.

Summarizing the research done so far, the method of directly measuring the pose of an output link using a laser, theodolite, and a horizontal level was presented. This method has the advantage of directly measuring the pose of the output link, which is the target of calibration, but it is impossible to compensate the line itself with sensors installed outside. The purpose of measuring real motion is to compare the theoretical and actual values of the kinematic model. Therefore, it is conceivable to find an error between the theoretical value and the set value of the kinematic model by reading the momentum of the active joint whenever the output motion of the mechanism is constrained and the constraint is satisfied. In order to generate the restraint motion, the restraint method using a single plane and digital indicator is applied to the parallel machine tool. However, since the motion of the output link is constrained in one plane, the observation is low, and since the correction data is taken in the limited plane, there is a concern of local correction. Therefore, the method to improve the observability by finding the optimal correction shape in the limited space should be presented. For the study of the optimal shape, Borm and Menq presented the observations and maximized the results. The simulation results show that the correction shape has a greater effect than the number of correction data. Recently, Benard and Khalil have applied each calibration system to Stewart-Gough's parallel instrument robots, suggesting variables that cannot be found in the calibration system and explaining why.

This study examines the observability of the constrained motion using one constraint plane and three or one digital indicator, and proposes a method to implement the overall correction by arbitrarily adjusting the height of the constraint plane. QR-separation of the observation matrix finds the optimal correction shape that can maximize the independence of the row vector and suggests the number of variables that can be observed from the rank of the observation matrix obtained from them.

In addition, the CAD / CAM function is applied to the movement program developed by the GUI (Graphic User Interface) method to apply the developed complex shape processing machine to the machine tool to enable the 3D complex shape processing work.

1. Compensation system by constraint operator

The most basic step of the calibration is to find the error between the theoretical value calculated from the kinematic model and the actual measured value. In this study, correction data are taken from the restraint motion. The distal end poses arbitrarily in unconstrained spaces but is restricted in confined spaces. Twist coordinates are introduced to define the constraint motion. The motion of all objects in space can be represented by rotation and transport around the twist axis. If we represent this as a 6 × 1 twisted coordinate vector,

Furnace ( T ₁ , T ₂ , T ₃ ) and ( T ₄ , T ₅ , T ₆ ) are the rotational and feed velocity components about the X, Y, and Z axes, respectively. Constrained motion takes all or part of the above six coordinates, sets the momentum on that coordinate and moves the output link to fit the constraint. To express this mathematically, we define the constraint operator C [.]. C constrains the motion between the two shapes contained in [.]. When shape x ^a and x ^b are a function of the kinematic variable vector ρ and the active joint value △ q ,

to be. From the restraint motion, the following correction equation is derived.

Where Δ Q = [Δ q ^{0 → 1} ,... ,? Q ^{el? E} ] ^T represents a change in? Q for generating e correction shapes, and ∑ = [ N ^l ,. , N ^e ] ^T represents the constraint momentum. If the design value of the kinematic variable is ρ 0, and the actual value and the error △ ρ , the following error model is derived.

Where ΔH is the error between the actual system and the kinematic model, J is the observation matrix of (r × p), p is the number of kinematic variables and r is the number of correction equations. Where r>> p When r = k * e. The matrix J is obtained by numerical analysis, which is obtained by the change of restraint momentum due to the small change in kinematic variables ε = 10 ^-6 .

2. Observability of Kinematic Variables under Restraint Motion

There are some kinematic variables that do not participate in motion or are bound together with other variables by constraints. These variables will not be found in the assumed constraints. Therefore, it is very important to find a correction shape that can contribute to the correction work as much as possible. In this case, each time the correction data is selected using the QR-separation method, the correction data is taken by finding an optimal shape to obtain a row that is most independent of the row vector of the observation matrix thus far obtained.

Compensation shape in Define a free vector, F, of unconstrained coordinates when

The vector F has a degree of freedom by the number of free coordinates and changes its size and direction to form a correction shape. The row vectors of the observation matrix obtained from the constraint motions of the two shapes

The row vectors obtained so far Independence from increases the rank of the observation matrix. To ensure independence, the shape-changing vector F must be different from the vectors created so far, and the greater the differentiation, the more information needed to calibrate to improve observation. shape The observation matrix by listing the row vectors obtained from

Obtain and QR-separate Ji + 1T. In this case, if the value of the j-th diagonal element of the R matrix is r _jj , the numerical values representing the independence of each row from it are expressed as follows.

Therefore, the shape that can minimize M (F) value is the best shape Becomes If the value of M (F) is optimized up to n-corrected shapes, the rank of the observation matrix is Rank ( J _n ) = n * k. If the rank is equal to the number of kinematic variables, then all variables are observable, but if they are small there are variables that cannot be observed.

In general, calibration requires more than twice the equation of kinematic parameters. Therefore, we need to add another shape to the shape obtained from the optimization work. In this study, the row vector obtained by the shape to be added Singular value of observation matrix including The concept of condition numbers is the ratio of the maximum and minimum values. The ideal case is to have a uniform eigenvalue for all eigenvectors, with a state number of one. Therefore, the added optimization shape is the number of states

Determine to minimize.

3. Calibration of parallel machine tool

Since the parallel mechanisms are assembled in space, the assembly tolerances are large and the number of kinematic parameters to be corrected is large. The observability of the restraint motion is examined by applying the principle of optimization of the correction shape to the parallel machine-type machine tool (4) (PMC, Parallel-typed Machining Center). The PMC is a linear actuator installed between the base, platform, and LA_i ( = 1, 2,... , 6). The upper end of LA_i is connected to point B _i on the base by a ball socket joint and the lower end is connected to point P _i on the platform by a universal joint. The shape of the PMC is determined by the (6 × 1) vector q representing the length of LA_i.

here

Δ q _i is the measurable momentum and q _{off, i} is the initial offset value and should be found from the calibration operation. The world coordinate system {W}, which is the basis of the correction work, is given to the constraint plane, and the coordinate systems {B} and {P} are fixed to the base and the platform, respectively. The X _W and Y _W axes of coordinate {W} are on the constraint plane and their direction is determined by the user. Point B _l is the origin of coordinate system {B} and the orientation of {B} matches the orientation of {W}. Also, point P _l is the origin of coordinate system {P} and the X _P axis Parallel to, point P ₁ , P ₂ , P ₆ on the XPYP plane. From the above definition, the following kinematic variables are zero.

Where ⁱ P _j represents the coordinate value of point j with respect to coordinate system {i}. If the U-joint and S-joint are perfect, the pose of {P} with respect to {W} can be represented by 36 kinematic variables. Kinematic variable vector

It is defined as Since the workpiece is installed on the surface plate and processed with the squareness, the top view, etc., the point B _i must be represented by the coordinate system {W} to correct the PMC for the surface plate.

Constraint planes and digital indicators can be used to create various constraints. Here, we examine the observability of 1) using 3 digital indicators and 2) using only one digital indicator. Place the table below the platform and install three digital indicators on the platform at equal intervals of 120 degrees. The points on the base are corrected based on the table, so the table does not have to be placed side by side on the base. The change in contact distance is shown as a digital value while the contact ball of the indicator is in contact with the surface of the surface. The 6-coordinate represents the output mechanism ^(W Digital values of Ω _E, ^W P _E) of the ^(W _Ex Ω, Ω ^W _{Ey, Ez} ^W P) changes, the indicator is changed. Therefore, if the shape is changed without any change in the three digital values, the constrained motion that occurs is represented by the constraint operator, C ₃ = diag (1, 1, 0, 0, 0, 1). Therefore, the three coordinates ^W Ω _Px , ^W Ω _Py , ^{and W} P _Pz represented by the coordinate system {P} are constrained. Since platform motion is constrained by one surface plane, there is a risk of local correction. However, the total correction is possible by adjusting the height of the constraint plane by arbitrarily changing the matrix of {E} for {P}.

shape Wow In constraint motion, the observability depends on the degree of freedom in which the vector F can vary. If the direction of the vector F is fixed and only the size is changed, the space for differentiating the shape is small and M (F) is immediately emitted. Active joint △ qi-> i + 1 changes ^W P _Px and ^W P _Py (i = 1,…, 6) was found. Each shape is changing to maximize its differentiation from previous shapes. We found six optimal shapes, but the rank is only 17 because the 18th diagonal element of matrix R is zero. Thus, 19 variables are unobservable. Add the degrees of freedom to the vector and find the optimized shape in plane and roll space with F = [ ^W P _Px , ^W P _Py , ^W Ω _Pz ] ^T , and then set the initial coordinates ( ^W Ω _Px , ^W Ω _Py , ^W P _Pz). The optimal shape that can be changed while maintaining) is nine. By changing the pose of the platform and changing the coordinates, the optimization can be increased to 11. Thus, under the constraint operator of C ₃ , up to 33 ranks cannot be found for three kinematic variables. When QR-separating the observation matrix by the correction shape, the 34th, 35th, 36th diagonal elements of the R-matrix are zero, ^{and W} P _{B 1 x} , ^W P _{B 1 y} , ^{and W} P _{B 1 z} cannot be observed among the kinematic variables. Do. This variable represents the position between {W} and {B} and does not participate in the relative constraint motion between poses so that the corresponding row vector becomes a zero vector. However, PMC does not need the absolute position between {W} and {B} by giving a relative pose command from the reference position.

_Assume the operator C ₁ = [0 0 0 0 0 1] that constrains the coordinate system ^W P _Ez using only one indicator. The origin of the coordinate system {E} is placed in the contact ball and the orientation is equal to the orientation of {P}. Since the restraint motion includes the position of the contact ball, add it to the correction variable by defining it as additional variables ρ a = [ ^P P _Ex , ^P P _Ey , ^P P _Ez ] ^T. Constrained operator C ₁ Wow The motion between the vectors F = [ ^W P _Ex , ^W P _Ey , ^W Ω _Ex , ^W Ω _Ey , ^W Ω _Ez ] ^T has a large free space to create the shape. The optimal shape can be obtained by obtaining up to 35 shapes and having up to 35 ranks. QR-separation of these observation matrices makes it impossible to observe ^W P _{B 1 x} , ^W P _{B 1 y} , ^W P _{B 1 z} , and ^B P _{B 2 y} among 39 variables including ρ a. Therefore, the position vectors ^W P _{B 1 x} , ^W P _{B 1 y} , ^{and W} P _{B 1 z} are set arbitrarily and define coordinates {B} to reduce the number of variables. In other words, the plane XBYB is parallel to the XWYW plane, and the point projected perpendicularly to the plane at point B ₂ is b ₂ and XB is Matches From the above definition

Becomes Therefore, one variable is reduced so that all variables can be observed. However, the coordinate system {B} does not coincide with {W} and is rotated at an angle with respect to the ZW axis. This result is because the height of the contact ball alone cannot obtain the coordinate values for the XW and YW axes. Therefore, the machining work for XW, YW axis set by the user is impossible. However, since the directions of ZB and ZW coincide, there is no problem in matching the verticality and flatness of the workpiece installed on the surface table.

The calibration data obtained from the optimal shape is not enough, so look for additional features that optimize the number of states. The number of states no longer decreases if the correction geometry provides more than four times the number of variables.

The simulation is carried out by the proposed method. Dividing the work area of the PMC of 3 constraining plane, and the C ₃ from the respective operator restraint flat shape 20, C ₁ constraining the operator to find the 60 shape to obtain a total of 180 correction formula.

The initial value is a design variable, and the error of variables q _{off and i} is ± 5mm, and the error of the remaining variables is ± 3mm. As a result, the cost function converges sufficiently to zero, and the average error of q _{off, i} is reduced to 3 μm. The same applies to the PMCs actually produced as they are applied to the simulation. The top view of the surface table (500 × 500 mm2) is 5µm, and the precision of the digital indicator is 1µm, which can measure up to 25mm displacement. The calibration data automatic acquisition device was used to measure the calibration data. It's not as fast as a simulation, but it's constantly converging to zero. The correction data taken by the constraint operators C ₃ and C ₁ are different _, but all variables converge to the same value within 20 μm, such as q _{off and 1} . This is a reliable basis for the proposed calibration system. However, since the cost function converges later than the simulation, it can be seen that the actual calibration data is less complete than the simulation data for the kinematic model.

To check the performance of the proposed calibration system, the precision of the instrument through the machining process was analyzed. The workpiece was machined with a multi-task machine and made of aluminum. The tool used a 12mm flat end mill to process linear and circular machining in the x- and y-directions.

4. CAD / CAM system development

The development of CAD / CAM system is a key technology for the creation of part program in GUI (Graphic User Interface) method. For this purpose, DXF file format was used for data exchange between general CAD drawing and CAM system under development, and Visual C ++ was used to increase scalability based on PC. The developed CAD / CAM program was implemented in two forms. It is developed for the existing users who are familiar with G-Code input method in the field, and the machining commands are automatically edited in the left edit window by selecting an object of drawing information read from CAD. You may. Developed for users familiar with the Windows environment, it is designed to speed up the creation of the part program by displaying the icon corresponding to the command instead of G-Code. In addition, the specialized part program has been functionalized and developed in-house to enhance the scalability of various tasks.

The observability of the correction system using the constraint operator was examined. Using QR-separation, we find the shape that maximizes the independence of the observation matrix and check the rank of the observation matrix obtained from them, and present the number of kinematic variables that can be observed. Define a numerical value representing the independence of the observation matrix and predict the observability according to the number of constraint coordinates from the rate of increase. When applied to parallel machine tools, the number of observations was 33 and all the necessary variables were observed when the platform was constrained by plane and roll motion with one plane and three digital indicators. If only one indicator was used, it was possible to calibrate by obtaining 35 observations including the installation position of the indicator. Non-observable variables are base coordinate values for the constraint plane and are not related to the relative constraint motion. Since these variables do not affect the calibration results, it is proved that the height of the constraint plane can be arbitrarily adjusted for the overall correction. We proposed a method to obtain a corrected shape sufficiently by adding a shape that minimizes the number of conditions of the observation matrix to the optimum corrected shape. As a result of applying this to the actual system, the calibration was successful. Even calibration data taken under different constraint operators proved the reliability of the proposed calibration system by converging kinematic parameters to the same value. There is still a relative error of less than 20 µm, but this is due to non-kinematic factors (backlash, measurement noise, changes, etc.) that the instrument has rather than originating from the calibration system. However, the automatic restraint motion generator minimized backlash and the precision digital indicator and the surface table minimized the non-kinematic factors that are always raised in the calibration.

Through this study, the correction by single plane eliminated the concern of low observability and local correction. Therefore, it showed that precise calibration is possible with only a simple digital indicator and a table.

The operation program of the complex shape processing machine developed for the machine tool was developed by the PC-based GUI (Graphic User Interface) method to facilitate the development and generation of the machining program. In particular, the processing instruction generation has been developed in two forms so that existing users and users familiar with Windows can use it without reluctance. This is the implementation of two-dimensional shape modeling, the development of the CAM function of the three-dimensional drawings in the future is a challenge.

Claims (1)

Parallel instrument type machine tool for complex machining, which is characterized by the method of reviewing the observability of the constrained motion using one constrained plane and three or one digital indicator, and implementing the overall correction by arbitrarily adjusting the constrained plane height. .