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英文原文
Research on a Novel Parallel Engraving Machine
and its Key Technologies
Abstract: In order to compensate the disadvantages of conventional engraving machine and exert the advantages of parallel mechanism, a novel parallel engraving machine is presented and some key technologies are studied in this paper. Mechanism performances are analyzed in terms of the first and the second order influence coefficient matrix firstly. So the sizes of mechanism, which are better for all the performance indices of both kinematics and dynamics, can be confirmed and the restriction due to considering only the first order influence coefficient matrix in the past is broken through. Therefore, the theory basis for designing the mechanism size of novel engraving machine with better performances is provided. In addition, method for tool path planning and control technology for engraving force is also studied in the paper. The proposed algorithm for tool path planning on curved surface can be applied to arbitrary spacial curved surface in theory, control technology for engraving force based on fuzzy neural network (FNN) has well adaptability to the changing environment. Research on teleoperation for parallel engraving machine based on B / S architecture resolves the key problems such as control mode, sharing mechanism for multiuser, real-time control for engraving job and real-time transmission for video information. Simulation results further show the feasibility and validity of the proposed methods.
Keywords: parallel mechanism, engraving machine, influence coefficient, performance indices, tool path planning, force control, fuzzy neural network, teleoperation
1 Introduction
Conventional computer engraving machine has played an important role in industries such as machinery machining, printing and dyeing and entertainment, but it has the inherent disadvantages such as cutting tool can be fed only along the fixed guideway, lower degree-of-freedom (DOF) of cutting tool, lower flexibility and mobility for machining etc. Parallel mechanism has the merits such as high mechanical stiffness, high load capacity, high precision, good dynamic performance etc (Zhen, H.; Ling-fu, K. & Yue-fa, F., 1997). According to the characteristics of parallel mechanism, it has been a hot research topic to apply parallel mechanism to the domain of future machining. By applying parallel mechanism to engraving domain, its inherent advantages can be fully exerted and the disadvantages of conventional engraving machine can be overcome or compensated. But as the special structure of parallel mechanism, the related theory and technology during its engraving is very different from that of conventional engraving machine, and it is a undeveloped research topic by now. In addition, with the development of computer network technology, the new concept and method such as network machining and manufacturing has become hot research topic (GQ, Huang & K.L,, Mak., 2001; Taylor, K. & Dalton, B., 2000; Ying-xue, Y. & Yong, L., 1999). A novel parallel engraving machine with six-axis linkage is proposed in this paper, which uses the 6-PUS parallel mechanism with 6-DOF as the prototype, and some key technologies such as size design, tool path planning, engraving force control and teleoperation are studied on this basis.
2. Confirming of mechanism type and engraving machine's size
2.1 Selection of mechanism and coordinate system
The selection of mechanism type is the first step for designing novel engraving machine, the following reasons make us select the 6-PUS parallel mechanism for designing our engraving machine. Comparing with traditional mechanism, 6-PUS parallel mechanism uses base platform, three uprights layout and high rigidity framework structure and has the merits such as high modularization, high accuracy and low cost. Itsmodel is shown in Fig.1.
Fig. 1. The model of 6-PUS parallel mechanism
As shown in Fig.1, 6-PUS parallel mechanism consists of base platform, dynamic platform and 6
branch chains with same structure, every branch joins with base platform through prismatic pairs
(P), slider of prismatic pairs joins with up end of the fixed length link through universal joint (U),
down end of the fixed length link joins with dynamic platform through sphere hinge (S), so it is
called 6-PUS parallel mechanism. The coordinate system of 6-PUS parallel engraving mechanism
is shown in Fig. 2. In Fig.2, the geometry centers of base platform and dynamic platform plane are
supposed as OB and op respectively. In every branch, the centers of prismatic pairs, universal joint
and sphere hinge are marked with Ai, Bi,, and Ci (i = 1,2, ..., 6) respectively. Coordinate system
OB-XBYBZB is fixed on base platform, taking {B} as briefly. The origin of {B} lies on geometry
center of base platform's up plane, axis ZB is vertical with base platform and directs to up, axis
YB directs to angle bisector of the first and second branch lead screw center line, and axis XB can
be determined with right-hand rule. Supposing the coordinate system set on dynamic platform is
op-xpypzp, taking {P} as briefly, its origin lies on geometry center of dynamic platform, the initial
state of dynamic platform system is consistent with that of base platform system completely.
Supposing the coordinate of op is (0,0, Z) in {B}, this configuration without relative rotation to
every axis is the initial configuration of this mechanism, and Z changing with mechanism's size.
On the basis of coordinate system mentioned, we use influence coefficient theory and the actual
parameters of this mechanism to calculate the first and the second order influence coefficient
matrix of every branch under different configuration. Then, we can get the first and the second
order integrated influence coefficient matrix H of the whole mechanism.
和
The significance and detailed solution process for influence coefficient matrix is omitted here, for more information please refer (Zhen, H.; Ling-fu, K. & Yue-fa, F., 1997).
Fig. 2. Coordinate system of 6-PUS parallel engraving mechanism
2.2 Mechanism performance analysis based on influence coefficient matrix
The performance of engraving machine will change with its size. To find out the better size for all the performance indices of both kinematics and dynamics, we obtain a group of mechanisms by changing its parameters. These mechanisms' length of fixed length links (L) range between 45cm and 55cm (step is 1cm), radius of dynamic platform (R) range between 10cm and 20cm (Step is 1cm). Other parameters of the mechanism is unchanging, so we get 121 mechanisms totally. Taking these mechanisms as research object, we confirm the sample point for every mechanism in its workspace with algorithm PerformanceAnalysis, then calculate the first and the second order influence coefficient matrix in every point. Furthermore, calculate all the performance indices in every sample point and draw all the global performance atlas of 121 mechanisms ultimately. To describe conveniently, we abbreviate the first and the second order integrated influence coefficient matrix Hq to G and H, and use Gω, Hω and Gυ, Hυ as the angular velocity submatrix and linear velocity submatrix of the first and the second order integrated influence coefficient matrix respectively, namely, We can change mechanism's parameters and adjust variable's step in the algorithm PerformanceAnalysis to meet actual analysis. The algorithm is programmed with MATLAB and the global performance atlas of 6-PUS mechanism are drawn (see Fig. 3 to Fig. 8), then the mechanism's performance is analyzed using the atlas. Table 1 shows the results of sample point number (abbr. to SPN) for 121 mechanisms respectively, the fixed link length of mechanism with sequence number (abbr. to SN) 1 is 45cm, its radius of dynamic platform is 10cm, the fixed link length of mechanism with SN 121 is 55cm, its radium of dynamic platform is 20cm, the rest may be deduced by analogy. In addition, table 2 gives the performance indices of some mechanism only, where the mean of SN is same as in table 1.
Description for algorithm PerformanceAnalysis:
PerformanceAnalysis Begin
For L = 45 To 55 / / scope of fixed length link
For R = 10 To 20 / / scope of radius of dynamic platform
SamplePointNumber = 0; / / initialization sample point number is zero for every mechanism
For x =-Maximum To + Maximum moving along Axis X Step 4cm
For y =-Maximum To + Maximum moving along Axis Y Step 4cm
For z =-Maximum To + Maximum moving along Axis Z Step 4cm
For α =-Maximum To + Maximum rotating around Axis X Step 12 °
For β =-Maximum To + Maximum rotating around Axis Y Step 12 °
For γ =-Maximum To + Maximum rotating around Axis Z Step 12 °
If sample point (x, y, z, α, β, γ)? Reachable point of mechanism's
workspace
Calculating the first order influence coefficient matrix and
its Frobenius norm at current point;
If The first order influence coefficient matrix is ??not
singular
SamplePointNumber = SamplePointNumber +1;
Calculating the second order influence
coefficient matrix and its Frobenius norm
calculating condition number at this point with
formula and accumulating sum of performance
indices;
/ / detailed formula is given in the following
of this section
Endif
Endif
Endfor
Endfor
Endfor
Endfor
Endfor
Endfor
Calculating all the performance indices of the mechanism at current size and append the results to corresponding data files for different performance index;
/ / performance index of the mechanism =(accumulating sum of performance indices at all sample points) / SamplePointNumber
/ / There are six data files for performance indices totally: angular velocity, linear velocity,angular acceleration, linear acceleration, force and moment, inertia force
Endfor
Endfor
Drawing all the global performance atlas of 6-PUS mechanism by all the index data files
(Every data file includes the information of 121 mechanisms);
/ / There are six performances atlas totally: angular velocity, linear velocity, angular acceleration, linear acceleration, force and moment, inertia force
End
Table 1. The SPN of 121 mechanisms in experiment
SN
SPN
六個(gè)性能指標(biāo)
角速度
線速度
角加速度
線加速度
力和力矩
慣性力
1
30962
0.17276
0.17442
0.06236
0.11315
0.01521
0.37454
2
28074
0.18248
0.18171
0.08075
0.13276
0.01456
0.40421
3
25848
0.19128
0.18836
0.09932
0.15184
0.01396
0.43136
4
23252
0.20087
0.19545
0.11897
0.17225
0.01348
0.46030
...
...
...
...
...
...
...
...
59
42390
0.21105
0.18995
0.10050
0.01304
0.01304
0.40233
60
37410
0.21915
0.19537
0.11308
0.17355
0.01257
0.42606
61
32446
0.22717
0.20041
0.12312
0.19230
0.01216
0.44929
...
...
...
...
...
...
...
...
119
28942
0.25779
0.20680
0.12265
0.22596
0.01064
0.47030
120
23998
0.26786
0.21185
0.12116
0.24139
0.01041
0.49500
121
19828
0.27714
0.21610
0.11399
0.25527
0.01017
0.51745
Table 2. Six performance indices of some mechanisms
2.2.1 Analysis of kinematics performance indices
2.2.1.1 Global performance indices of angular velocity and linear velocity
As the influence coefficient G of engraving mechanism is not a constant matrix, it makes the measuring index for parallel mechanism based on G not to be a constant matrix also, so we can't utilize one value to measure the good or bad of the dexterity, isotropy and controlling accuracy (Xi-juan, G., 2002). Here, we define parallel mechanism global performance indices of angular velocity and linear velocity as following respectively
Where W is the reachable workspace of mechanism,
anddenote the condition numbers for angular velocity and linear velocity respectively (Where | | · | | denotes Frobenius norm of matrix, superscript '+' denotes generalized inverse matrix, the same mean as following). We can get the performance indices' value of the angular velocity and linear velocity according to the condition numbers of every mechanism's sample points. Replacing the underlined part in algorithm PerformanceAnalysis with two formulas in (1) respectively, we can draw the performance atlas for angular velocity and linear velocity as shown in Fig.3 and fig.4 based on 121 mechanisms' indices values ??of angular velocity and linear velocity. According to the rule that the bigger ηJ (J ∈ {Gω, Gv}), the higher dexterity and controlling accuracy of the mechanism, from Fig.3 we can see that the mechanism performance index of angular velocity is not changing with the link length when the changing range of R is not big, but it has the trend that the bigger R, the better
Fig. 3. Atlas of angular velocity global performance
Fig. 4. Atlas of linear velocity global performance
performance index of angular velocity, furthermore, the index of mechanism angular velocity is better when L = 46.5cm ~ 49.5cm and R = 19.5cm, namely, the output error of angular velocity is smaller. Similarly, from Fig.4 we know that the mechanism index of linear velocity is better when L = 45cm ~ 48cm and R = 19cm, that is to say,the output error of linear velocity is smaller.
2.2.1.2 Global performance indices of angular acceleration and linear acceleration.Considering the influences on acceleration of both the first and the second order influence coefficient matrix, the condition numbers of angular acceleration and linear acceleration for 6-DOF parallel mechanism
are (Xi-juan, G., 2002; Xi-juan, G. & Zhen, H., 2002)
Where, a and b is error coefficient.So the global performance indices of angularacceleration and
linear acceleration for parallelengraving mechanism can be defined as
Where Supposed the mechanism error is smaller than 2%
(that is, a = b = 0.02), replacing the underlined part in algorithm .PerformanceAnalysis with formula (4), we can draw the performance atlas for angular acceleration and linear acceleration as shown in Fig.5 and Fig.6. As same as the evaluating method for velocity performance index, from Fig. 5 we can see that the angle acceleration performance of mechanism is better when nearly L =
45cm ~ 47cm and R = 16cm ~ 20cm, output error is smaller accordingly. Among the 121
mechanism we studied, its maximum is 0.16399.
Fig.5. Atlas of angular acceleration global performance
By observing Fig.6 carefully, we know that performance
of linear acceleration is better when nearly L=45cm~48cm and R=19.5cm, accordingly, output error should be smaller. From above analysis, we know that mechanism size with good indices for linear velocity and linear acceleration is coincidence in some degree among the 121 mechanisms we studied, but performance index of angular velocity and angular acceleration may not the best in the same size, so it can’t get coincidence. Thus, our analysis will be helpful for designing and choosing mechanism by actual needs. Similarly, analyzing method of kinematics performance indices is the same when other parameters of the mechanism are changed.
Fig. 6 . Atlas of linear acceleration global performance
2.2.2 Analysis of dynamics performance indices
2.2.2.1 Analysis of power and moment performance
Index. The condition number of power and moment performance index based on the first order influence coefficient matrix of power GF for 6-DOF parallel mechanism can be defined as(Xi-juan,G.,2002)
Similarly, we define global performance index of power and moment for 6-DOF parallel mechanism as
We suppose that power and moment of parallel mechanism is isotropy when ηJ=1. With formula (5) as condition number, replacing the underlined part in algorithm with formula (6), we can draw the performance atlas for power and moment as shown in Fig.7. From Fig. 7 we can see in the size range of our experiment the performance index for power and moment would have little change with the link length when the radius of dynamic platform is less then 14cm. The performanc index for mechanism’s power and moment will be bigger when L=45cm~46cm and radius of dynamic platform R=10cm,here, performance of power and moment will be better.
Fig. 7. Atlas of global performance of force and moment
2.2.2.2 Analysis of inertia force performance index
Considering both the first and the second order influence coefficient matrix, the condition number of inertia force for 6-DOF parallel mechanism is defined as(Xi-juan,G.,2002)
Where [Gω ]i is the ith column of matrixGω , i=1,2,3.Then global performance index of engraving
mechanism’s inertia force can be defined as
Obviously, the bigger value of η G+H , the smaller inertia force of mechanism and the higher controlling accuracy. Replacing the underlined part in algorithm PerformanceAnalysis with formula(8), we can draw the performance atlas for inertia force as shown in Fig.8. According to the rule that the bigger value of η G+H , the smaller inertia force of mechanism and the higher controlling accuracy, by observing Fig. 8 carefully, we can see that the inertia force performance index of mechanism is getting better when the link length is getting longer and radius of dynamic platform is getting bigger. Furthermore, the inertia force performance index of mechanism will be the best when nearly L=45cm~48cm and R=19.5cm, that is to say, the inertia force performance of mechanism is the best, inertia force is the smallest, sensitivity is the best and dexterity is the highest. According to discusses above, we draw a conclusion that mechanism size with good performance index for power and moment and inertia force is coincidence not in all the time. This result indicates thatthe traditional designing and choosing mechanism method based on only the first order influence coefficient exists some restriction, we have to choose mechanism size on the basis of our analysis and actual demands.
Fig. 8. Atlas of global performance of inertia force
2.3 The results of size design and summary
Summarizing previous analysis, we know that 6-PUS robot mechanism’s all performance indices are better except force and moment when L=45cm~47cm and R=19cm, the actual size of mechanism with this type owned by laboratory is at the above-mentioned scope. We also find that its performances are same with the results of theory analysis by running the mechanism in deed, so prove the correctness of our theory. To validate our theory analysis further, we also do lots of simulations for this mechanism with other sizes. The results are same with those of theory analysis, so we can draw the conclusion that, in a generally way, there is not a mechanism whose all indices are better at the same time for both the kinematics and dynamics. In fact, we can only select the mechanism with relative better performance for all the indices to meet our need. On the basis for considering all the performance indices, the final sizes of novel parallel engraving machine that we designed are following: The length of fixed link L is 46cm, the radius of dynamic platform R is 19cm,the radius of base platform is 38cm38cm,{?P1,?P2,?P3,?P4,?P5,?P6}={45°,135°,165°,255
°,285°,15°},{φB1,φB2,φB3,φB4,φB5,φB6 }={82°,97°,202°,217°,322°,337°}, ?C=30°,φA=15°. Where ?Pi (i=1，2，…，6)is the angle between tieline from dynamic platform’s center op to Ci and the axis xp’s positive direction of dynamic platform’s coordinate system, φBi (i=1，2，…，6)is the angle between tieline from base platform’s center OB to Ai and the axis XB’s positive direction of base platform’s coordinate system. ?C and φA is the smaller central angle of hexagon made by Ci and Ai(i=1，2，…，6) respectively.
3. Research on path planning of engraving
A series of engraving paths can be obtained after the im
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