CHINESE JOURNAL OF MECHANICAL ENGINEERING ·930·

Vol. 29,aNo. 5,a2016

DOI: 10.3901/CJME.2016.0425.059, available online at www.springerlink.com; www.cjmenet.com

Design and Accuracy Analysis of a Metamorphic CNC Flame Cutting Machine for Ship Manufacturing HU Shenghai, ZHANG Manhui*, ZHANG Baoping, CHEN Xi, and YU Wei College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China Received December 9, 2015; revised March 14, 2016; accepted April 25, 2016

Abstract: The current research of processing large size fabrication holes on complex spatial curved surface mainly focuses on the CNC flame cutting machines design for ship hull of ship manufacturing. However, the existing machines cannot meet the continuous cutting requirements with variable pass conditions through their fixed configuration, and cannot realize high-precision processing as the accuracy theory is not studied adequately. This paper deals with structure design and accuracy prediction technology of novel machine tools for solving the problem of continuous and high-precision cutting. The needed variable trajectory and variable pose kinematic characteristics of non-contact cutting tool are figured out and a metamorphic CNC flame cutting machine designed through metamorphic principle is presented. To analyze kinematic accuracy of the machine, models of joint clearances, manufacturing tolerances and errors in the input variables and error models considering the combined effects are derived based on screw theory after establishing ideal kinematic models. Numerical simulations, processing experiment and trajectory tracking experiment are conducted relative to an eccentric hole with bevels on cylindrical surface respectively. The results of cutting pass contour and kinematic error interval which the position error is from –0.975 mm to +0.628 mm and orientation error is from –0.01 rad to +0.01 rad indicate that the developed machine can complete cutting process continuously and effectively, and the established kinematic error models are effective although the interval is within a ‘large’ range. It also shows the matching property between metamorphic principle and variable working tasks, and the mapping correlation between original designing parameters and kinematic errors of machines. This research develops a metamorphic CNC flame cutting machine and establishes kinematic error models for accuracy analysis of machine tools. Keywords: CNC cutting machine, metamorphic principle, accuracy analysis, screw theory, error model

1



Introduction

During the manufacturing process of large hull structures of modern vessels those travel on or under the water surface, lots of holes with large size and complicated contours, as shown in Fig. 1, have to be cut on the spatial curved surfaces of hull. The processing of those holes characterized by variable altitudes, angles of bevels and the thickness of removed materials is a key problem during the manufacturing process. And the processing precision of those holes has direct influence on the quality of subsequent welding sequence, the tactical and weapon system qualities. The traditional cutting method follows the process of manual lofting, making prototype, and hand cutting and finally fitting regrinding[1]. However, due to the variation of hull structures and materials, traditional method cannot meet the requirements of precision and shorten the construction cycle of modern vessels. Therefore, it is important to study digital agile cutting technique[2–3] * Corresponding author. E-mail: [email protected] Supported by National Natural Science Foundation of China (Grant No. 51175099) © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2016

for ship manufacturing, and develop processing equipments for cutting holes on complex spatial curved surface of hull. Previous relative works, which have already been done by scholars, research institutes and so on, have developed kinds of CNC cutting machines. KOMATSUBARA, et al[4], developed a high speed CNC cutting machine by using linear motor and worked out the operation interface. The machine can move along X and Y axis and be adapted to the cutting process of flat plate. GORDON, et al[5], designed a cutting machine which can be used to process elliptical holes and simple holes on plane. WANG, et al[6], studied the pose control in intersecting curve with bevel at the end of pipe, and studied the selections of torch trajectory and pose during processing on spatial surfaces. SUN, et al[7], proposed a new pipe cutting equipment to replace the less-automated cutting machines. The equipment can process small size and simple curved surfaces with high precision. ESAB covered almost all kinds of non-contact cutting machines, and those can process on small size and simple surfaces with high precision[8]. Patents designed three kinds of cutting equipments[9–11], and they can place the workpiece on the equipment’s platform and process bevel with various shapes. WANG, et al[12], developed a large diameter CNC

CHINESE JOURNAL OF MECHANICAL ENGINEERING cutting machine which is the special equipment for cutting large holes with bevels on cylindrical shells. It can accomplish the cutting task of oblique jacks on curved surface. However, these machines have several limitations when they are used to process holes on complex spatial curved surface. Most of them are concerned with holes on miniature curved surfaces or flat plates. The special cutting equipment developed by the author[12] needs manual adjustment corresponding to different working conditions. So, it is extremely necessary to design a cutting equipment to satisfy the requirements of effectively cutting and shorten construction cycle. It means the machine needs to drive the cutting tools moving continuously in accordance with different working condition during a working cycle. In other words, the machine should have the ability to have its structure transformed from one kind to another to accomplish different tasks[13–14].

formula. YU, et al[24] and BRIOT, et al[25], studied the influences of active inputs on the end-effector kinematic accuracy by using geometric method. This work investigates the digital agile cutting technique for ship manufacturing, a novel CNC flame cutting machine for cutting large size holes on complex spatial curved surface of hull is developed and its kinematic accuracy analysis is done. The remainder of this paper is organized as follows: Section 2 figures out the needed variable trajectory and variable pose kinematic characteristics of non-contact cutting tool for cutting continuously and automatically. Section 3 presents the developed CNC flame cutting machine. Then the ideal kinematic models derived by using nominal parameters, the models of three types of error sources in kinematic parameters and kinematic error models of the machine are established respectively in section 4. Section 5 not only deals with the numerical simulations of the overall established models, but also covers trajectory tracking experiment and processing experiment for a cutting example. Finally, conclusions are given in section 6.

2

Fig. 1. Different kinds of ‘large holes’ on the spatial curved surface of hull

On the subject of kinematic accuracy analysis for the machine or mechanism, besides measuring through experiments[15–16], the kinematic accuracy can be also obtained by establishing and solving error models. In process of theoretical analysis, many researches considered the influences of clearances in pairs, manufacturing and assembly tolerances and errors of active inputs. TSAI, et al[17], considered the clearance as virtual links with the hypothesis of rigid bodies, which is the basic method for studying the influence of joint clearances. FRISOLI, et al[18], proposed the step-by-step procedure based on screw theory for the analysis of position accuracy in parallel mechanism with revolute joint clearances. HUANG, et al[19], transformed the D-H parameter errors into virtual kinematic pairs and expressed them as error screws. Then the error model is established to analyze kinematic accuracy. WU, et al[20–21], studied the combined influences of joint clearances and manufacturing tolerances by using interval analysis method. CHAKER, et al[22], also analyzed the combined influences of clearances and manufacturing tolerances, and concluded that the linear superposition principle is not suitable for accuracy analysis of end-effector. KUMARASWAMY, et al[23], treated the tolerance parameters as the active variables of virtual kinematic pairs. Then the kinematic error was obtained by establishing the actual kinematic models by using the POE

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Mathematical Models of Spatial Curved Surface of Hull and Inverse Kinematics of Non-contact Cutting Tool

As shown in Fig. 1, the complex spatial curved surfaces of hull can be divided into two categories. One is the “regular” that can be described and classified by extracting the feature parameters of the surface and described directly by parametric mathematical model. The other is “irregular” that the models can only be obtained by using geometric theory[26] and the technique of surface reconstruction. In this section, the first kind is concerned. 2.1

Parametric modeling of cutting surfaces and pass curves The surface of an ideal cylindrical shell is parametric modeled based on designated mapping method[27]. With the correlations between the mapping coordinates of front and back nodes, parametric models of the original curves and curved surfaces can be derived. And other kinds of ‘regular’ surfaces can also be dealt with in the same way. In rectangular coordinate system O-XYZ , the cross section of hole with upper and lower bevels on the cylindrical shell is shown in Fig. 2(a). Its projection on the OXY plane is selected as the mapping domain. 1) Curved surface models of external contour, internal contour and straight hole contour ì ì ï x12 + z12 = ( R + S ) 2 , ï x 2 + z2 2 = R 2 ,  2 : ïí 2 ï ï ï y1 = y1 , ï î î y2 = y2 ,

(1)

ìï( x - e) 2 + y 2 = r 2 , 3 3 ïï z3 = z3 , î

(2)

 1 : ïí

 3 : ïí

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HU Shenghai, et al: Design and Accuracy Analysis of a Metamorphic CNC Flame Cutting Machine for Ship Manufacturing

ì x51 = e + r cos  , x61 = e + l2 cos  , ï ï ï ï y52 = r sin  , y62 = l2 sin  , ï ï ï  5 6 : í ï z53 = Rs 2 2 - (e + r cos  ) 2 , ï ï ï 2 2 ï ï ï î z63 = R - (e + l2 cos  ) ,

where S is the thickness of cylindrical shell, R is the radius of internal surface, e is the eccentricity of the intersecting hole, r is the radius of projection of intersecting hole.

(5)

where l1 ( l2 ) is the parameter of the upper (lower) bevel wind-line on the mapping domain, Rs1 ( Rs 2 ) is the radius of the upper (lower) bevel root-cutting. 3) Models of upper and lower bevel angles in the intersecting hole By spatial geometry theory, the upper bevel angle e and the lower bevel angle  f are

ì ï ï E E • E1 E3 ues ï = cos e = 1 2 , ï ï E1 E2 • E1 E3 (6) (l1 - r )2 + ues 2 í ï ï ï 2 2 2 2 ï ï ï îues = ( R + S ) - (e + l1 c  ) - Rs1 - (e + r c  ) , ì ï ï u fs FF •FF ï ï cos  f = 1 2 1 3 = , ï ï F1 F2 • F1 F3 (l2 - r ) 2 + u fs 2 í ï ï ï 2 2 2 2 ï ï ï îu fs = R - (e + l2 c  ) - Rs 2 - (e + r c  ) ,

(7)

where c = cos and s = sin . The continuously varying of bevel angles is an important precondition of high quality welding process during the ship manufacturing. By smooth transition principle[28], trigonometric parameter model of bevel angle is Fig. 2.

Cross section of ‘large holes’ hole and its cutting process

2) Curve models of straight hole, upper and lower bevel contours in the intersecting hole A circle region on the OXY plane is chosen as the mapping domain and angle  is chosen as kinematic variable. Curves models of straight hole  1 and  2 are

ìï x11 = e + r cos  , x21 = e + r cos  , ïï ïï y12 = r sin  , y22 = r sin  , ï  1 2 : í ïï z13 = ( R + S ) 2 - (e + r cos  ) 2 , ïï ïï z = R 2 - (e + r cos  ) 2 , ïî 23

(3)

where the angular variable  Î [ 0, 2π ] , and radius r can be denotes as r = r ( ) . The models of upper bevel contours  3 ,  4 and lower bevel contours  5 ,  6 are

ïìï x31 = e + r cos  , x41 = e + l1 cos  , ïï y42 = l1 sin  , ïï y32 = r sin  ,  3 4 : ïí ïï z33 = Rs12 - (e + r cos  ) 2 , ïï ïï z = ( R + S ) 2 - (e + l cos  ) 2 , 1 ïî 43

(4)

é

i = Ki êê ë

ù 1 sin( M i )ú + Ni , ú Mi û

(8)

where i = e, f , and Ki , M i , Ni are undetermined constants. 2.2

Inverse solution of the pose and trajectory of non-contact cutting tool To meet the cutting requirements of high-precision and high-quality, the pose models need to combine the requirements of flame cutting as shown in Fig. 2(b). During the process of straight hole cutting, the cutting tool should offsets r0 along the radius direction of the straight hole contour and offsets s0 along the Z direction of  1 . Mathematical model of the end-points g1 and h1 of the tool axis can be expressed as ì xg1 = xh1 = e + (r - r0 ) cos  , ï ï ï ï ï ï y g1 = yh1 = (r - r0 ) sin  , g1h1 : ï (9) í ï z g1 = ( R + S ) 2 - [e + (r - r0 ) cos  ]2 + s0 , ï ï ï ï 2 2 ï ï î zh1 = ( R + S ) - [e + (r - r0 ) cos  ] + s0 + L,

where L is the length of cutting tool, r0 denotes the

CHINESE JOURNAL OF MECHANICAL ENGINEERING actual cylindrical radius of the cone torch corresponding to the cutting tool, s0 is the height from the cutting blowpipe to the cutting surface that corresponds to the non-contact distance. During the process of the upper bevel cutting, due to the varying of bevel angle, the cutting tool should offset s0 along the wind-line of  3 , and offset r0 along the vertical direction of the wind-line. The model of the cutting tool axis g3 h3 is derived as ïìï xg 3 = e + r c  - s0 s  e c  - r0 c  e c  , ïï ïï y g 3 = r s  - s0 s  e s  - r0 c  e s  , ïï ïï z = R 2 - (e + r c  ) 2 - s c  + r s  , s1 e e 0 0 g3 h3 : ïí g 3 ïï x = e + r c  - ( s + L) s  c  - r c  c  , e e 0 0 ïï h 3 ïï yh 3 = r s  - ( s0 + L) s  e s  - r0 c  e s  , ïï ïï z = R 2 - (e + r c  ) 2 - ( s + L) c  + r s  . s1 e e 0 0 ïî h 3

expression of cutting thickness can be derived from Eq. (3) to Eq. (5) as

ìïu = ( R + S )2 - (e + r c  )2 - R 2 - (e + r c  )2 , ïï 1 (14) í ïïu = (l - r )2 + u 2 , u3 = (l2 - r )2 + u fs 2 , es 1 ïî 2 where u1 , u2 and u3 are the cutting thickness of straight hole, upper and lower bevels, l1 = l1 (e ) and l2 = l2 ( f ) are the expressions of the radius on the planar mapping domain. According to the process parameters, plate material and the empirical cutting velocity that determined by cutting technology handbook[1] and experiments, the implicit expressions of velocity scalar relative to cutting thickness model can be derived through cubic polynomial fitting as ïìïv1 = v (u1 ), í ïîï1 = 0,

(10) Similarly, mathematical expression of axis g2 h2 during the process of the lower bevel cutting can be derived as ìï xg 2 = e + r c  - s0 s  f c  - r0 c  f c  , ïï ïï y = r s  - s s  s  - r c  s  , 0 f 0 f ïï g 2 ïï 2 2 ï z g 2 = Rs 2 - (e + r c  ) + s0 c  f - r0 s  f , g 2 h2 : ïí ïï xh 2 = e + r c  - ( s0 + L) s  f c  - r0 c  f c  , ïï ïï yh 2 = r s  - ( s0 + L) s  f s  - r0 c  f s  , ïï ïï z = R 2 - (e + r c  )2 + ( s + L) c  - r s  . s2 0 f 0 f ïî h 2 (11)

In rectangular coordinate O - XYZ , the pose of cutting tool can be expressed by the angles between the vector hi gi (i = 1, 2,3) and three axes. They can be expressed as ìï xi = arccos(( xgi - xhi ) / HGi ), ïï ïï = arccos(( y - y ) / HG ), gi hi i ïï yi (12) í = arccos(( z - z ) / HG ), ïï zi gi hi i ïï ïï HG = ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 . gi hi gi hi gi hi ïî i Then, the inverse solution for the pose of cutting tool can be derived as

·933·

ìïv2 = v (u2 ), ïí ïï2 = e , î

ïïìv3 = v (u3 ), í ïï3 =  f , î

(15)

where v j ( j = 1, 2,3) is the linear velocity scalar of cutting tool during cutting straight hole, upper and lower bevels,  j denotes angular velocity scalar between tool axis and OZ axis. Based on Eq. (3) to Eq. (5) and the Gram-Schmidt orthogonalization algorithm, expressions of the tangent vector of straight hole, upper and lower bevels holes’ curves can be obtained, which is the direction of linear velocity of the cutting tool in the rectangular coordinate. Therefore, the velocity inverse solutions of the cutting tool are obtained.

3

Structure Design and Analysis on Working Principles of a Metamorphic CNC Flame Cutting Machine

According to the inverse kinematic models of the cutting tool obtained in section 2, it can be figured out that the cutting tool should possess the features of varying pose and trajectory to meet the requirements of cutting large size holes with upper and lower bevels. Ignoring the processes of initialization and return-to-zero, its movement flow is shown in Fig. 3.

ì ï  x1 = π / 2;  x 2 = arccos(s  f c  );  x 3 = arccos(s e c  ), ï ï ï í y1 = π / 2;  y 2 = arccos(s  f s  );  y 3 = arccos(s  e s  ), ï ï ï  z 3 = e . ï ï î z1 = π;  z 2 =  - f ; (13) 2.3

Inverse solution of the velocity of non-contact cutting tool To ensure the quality of flame cutting, the velocity models should correspond to the cutting thickness. The

Fig. 3.

Movement flow of the cutting tool during a working cycle

As shown in Fig. 3, the “state” of the cutting tool changes with the working condition during a working cycle,

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HU Shenghai, et al: Design and Accuracy Analysis of a Metamorphic CNC Flame Cutting Machine for Ship Manufacturing

which indicates that the machine for driving the cutting tool should have the feature of changing its ‘state’. Thus a novel decoupled CNC flame cutting machine is designed based on metamorphic principle[13, 29], the material object picture and schematic diagram are shown in Fig. 4.

Table 1.

Analysis of topological structures for every working configuration of Fig. 4(b) Active joints number

Locked joints number

Straight hole cutting configuration

1, 2, 3

4, 5 6, 7 8

Pose adjusting configuration

2, 3 4, 6

1

Bevel cutting configuration

1, 2 3, 4

6

Configuration

Topological graphic

R

P P

P

P

P R

P P

P R

R

R R

R

R R

(1) Straight hole cutting configuration The mechanism adjusts the initial pose of the cutting tool through merging adjacent components by locking joint 4 to joint 8. The final configuration is shown in Fig. 5(a), where the joints represented by dashed lines are locked and joints packaged in double dot lines represent series structure on the front-end of the machine. Thus the swing assembly is equivalent to an integral component. The horizontal rotating platform, the radical adjusting module and the lifting body drive the cutting tool and meet the requirements of straight hole cutting. The second line of Table 1 gives the topological structure of current configuration. The number of degree of freedom F can be counted by[30] 3

F = å fi = R + P + P = 1 + 1 + 1 = 3 . Fig. 4. The designed CNC flame cutting machine 1. Horizontal rotating platform, 2. Radical adjusting module, 3. Lifting body, 4. Swing assembly, 5. Arc plate, 6. Swing bridge, 7. Upper platform, 8. Column, 9. Slider, 10. Electric group, 11. Adjustable frame

The virtual prototype and components of the machine are shown in Appendix A. Then, its equivalent mechanism shown in Fig. 4(b) is analyzed. Besides, {S} is machine coordinate system and {T } is tool coordinate system. S/i (i = 1, 2,,8) are the screws of joint axes and the related kinematic variable parameters are i (di ). c j ( j = 0, 1,,6) are structural parameters,  is the attaching parameter between virtual links and cutting tool. For each phase shown in Fig. 3, the mechanism changes its configuration by merging components[14] to adjust with different working conditions. The topological structures for every working configuration are shown in Table 1 and the operating process can be described in detail as follows. Note that, the joints number belongs to subscript of screws, R and P represent the type code of revolute and prismatic joint.

(16)

i =1

(2) Pose adjusting configuration By locking joint 1, the pose adjusting of cutting tool can be realized. It consists of two cases: from straight hole to lower bevel that shown in Fig. 5(b); and from lower bevel to upper bevel that shown in Fig. 5(d). In this configuration, the radical adjusting module, the lifting body and the swing assembly work together to convert the cutting tool from the straight hole configuration to the lower bevel initial configuration (or from the lower bevel final configuration to the upper bevel initial configuration), meeting the requirements of cutting continuously during a working cycle. The third line in Table.1 gives the topological structure, and the number of degree of freedom F can be counted by

F = g0 + b(n - g -1) + g = 2 + 3(5 - 5 -1) + 5 = 4,

(17)

where g0 is the mobility in open loop, b is the order of the closed loop, n is the number of links and g the number of joints in closed loop.

CHINESE JOURNAL OF MECHANICAL ENGINEERING

·935·

(3) Bevel cutting configuration The mechanism realizes bevel cutting by locking joint 6. Corresponding to different angles in joint 6, the lower bevel cutting configuration is shown in Fig. 5(c) and the upper bevel cutting configuration is shown in Fig. 5(e). In this configuration, the horizontal rotating platform, the radical adjusting module and lifting body drive the cutting tool moving. While the swing assembly is no longer an integral component, it changes the poses of cutting tool, meeting the requirements of continuously varying of the bevel angel. The fourth line in Table 1 gives the topological structure and number of degree of freedom F can be counted by

closed-loop structural equations and the processing task. 1 d 20 , 1d30 are initial values of the prismatic joints depending on the pass contour in the task, the geometric meaning of s0 is as same as that in Eq. (9), while its physical meaning is the spatial distance between origin /7 . point of {T } and ideal axis S Based on the ideal kinematic model in Eq. (19) and the screw expression of spatial Jacobian matrix[31], the ideal velocity mapping model of the end-effector in {S} can be expressed as

F = g0 + b(n - g -1) + g = 3 + 3(4 - 4 -1) + 4 = 4. (18)

where  = [1 d2 d3 ]T is the array of joint velocity, J st1 ( ) is the spatial Jacobian matrix of the straight hole cutting configuration, and its expression is

Vˆst1 = J st1 ( ) ,

J st1 ( ) = éë S/1

S/2¢

S/3¢ ùû .

(21)

(22)

Besides, elements in the matrix can be derived by

S/i¢ = Ad(exp(1S/ˆ1 ) exp(i-1S/ˆi-1 )) S/i . Fig. 5.

4

Configuration-complete of a working cycle

Kinematics and Accuracy Analysis of the Metamorphic CNC Flame Cutting Machine

4.1 Ideal kinematic model of the machine during a working cycle According to the topological structure and the corresponding schematic diagram of straight hole cutting configuration, the ideal forward kinematics equation can be derived based on the POE formula[31] as

gst1 ( ) = exp(1S/ˆ1 ) exp(d 2 S/ˆ2 ) exp(d3 S/ˆ3 ) gst1 (0) , (19)

/2 and S/3 are kinematic screws of ideal joint where S/1 , S axes. gst1 (0) denotes initial configuration and can be decomposed into the form of product of transformation matrixes

(23)

Assuming that p1 ( ) is the position components of the origin point of tool coordinate in {S}, velocity explicit expression can be obtained from the mapping relationship T v st1 = (Vˆst1 ) [ p1 ( ) 1] .

(24)

For the pose adjusting configurations shown in Figs. 5(b) and 5(d), the ideal forward kinematics equation can be derived as

gst 2 ( ) = exp(d 2 S/ˆ2 ) exp(d3 S/ˆ3 ) exp(6 S/ˆ6 ) exp(7 S/ˆ7 ) gst 2 (0) , (25) where 7 is the intermediate variable which is given in Appendix B, gst 2 (0) can also be obtained from Eq. (20) while the left superscript 1 need to be replaced by 2 and 4. For the pose adjusting configuration, the purpose is to adjust the pose of the cutting tool instead of actual processing. Thus velocity analysis is neglected in this part. The ideal forward kinematics equation of bevel cutting configuration shown in Figs. 5(c) and 5(e) is

ìï g (0) = Trans( X , 1t )Trans( Z , 1t )Rot(Y , 1 ) ´ x z 6 ïï st1 gst 3 ( ) = exp(1S/ˆ1 ) exp(d 2 S/ˆ2 ) exp(d3 S/ˆ3 ) exp(7 S/ˆ7 ) gst 3 (0) , ïï æ ö π 1 Rot ççY , -  +  7 ÷÷÷ Trans( Z , -s0 ), (26) ïïï çè 2 ø (20) í ïï 1 ïï t x = 1d 20 + c2 + c4 cos 1 6 , where gst 3 (0) is similar with gst 2 (0) and can be still ïï obtained from Eq. (20). ïïî 1t z = c0 - c1 - 1d30 - c3 - c4 sin 16 , The ideal velocity mapping relationship of bevel cutting configuration in {S} is where Trans(⋅) and Rot(⋅) correspond to translation and rotation matrixes. 16 , 17 are initial value for kinematic Vˆst 3 = J st 3 ( ) , (27) variables of the locked joints, which are derived from the

·936·

 = éêë1 d2

HU Shenghai, et al: Design and Accuracy Analysis of a Metamorphic CNC Flame Cutting Machine for Ship Manufacturing T d3 7 ùúû , J st 3 ( ) = éë S/1

S/2¢

S/3¢

S/7¢ ùû . (28)

Then the velocity explicit equation can be expressed as T

v st 3 = (Vˆst 3 ) [ p3 ( ) 1] .

(29)

The angular velocity model of bevel cutting configuration needs not to be established, as it can be obtained directly from the intermediate variable 37 . 4.2 Screw model of joint clearances In this section, clearance characteristic element represented the relative pose of two adjacent components and clearance spaces limited the movable range of the element are used to establish the clearance model. As shown in Fig. 6, the following assumptions are made.

where the subscript i3, i 6 denote the third and sixth term of the screw coordinate of actual axis respectively, ri is the vector of an arbitrary point on actual axis in the local coordinate, rd is the vector of an arbitrary point on the ideal axis in the local coordinate, di is the distance between the actual axis and the ideal axis, which is expressed as di = ri - rd . The screw coordinate of clearance characteristic element shown in Fig. 6(a) is S/rR = [ R

T

rR ´ R ] = [ S R1

SR2

SR3

SR4

T

SR6 ] ,

SR5

(32)

ì ï  (c  - c  2 ) L  (s  + s  2 ) ï ï S R1 = a 1 , SR4 = R a 1 , ï SR 2S R ï ï ï ï  a (s  1 - s  2 ) -LR  a (c  1 + c  2 ) ï , SR5 = , í SR2 = ï S 2S R R ï ï ï ï  2 s( 1 -  2 ) L ï , S R3 = R , SR6 = a ï ï SR SR ï ï î

(33)

where  a denotes the radical clearance of revolute joint,

LR is the fitting length, 1 ,  2 are the angle in xoy plane, variable S R = 2 a 2 [1 - cos( 1 -  2 )] + LR 2 . As the clearance model is established by kinematic relationship, the accurate angle at any moment can not be determined, thus it is assigned to be a random variable within [ 0, 2π ]. The screw coordinate of clearance characteristic element shown in Fig. 6(b) is Fig. 6.

T S/rP = [ 0 v P ] = [ 0 0 0 S P 4

Indication of clearances and two points of contact model of joints

(1) Ignoring the axial clearance of revolute joint and the radical clearance of prismatic joint in clearance space; (2) Clearance characteristic element and the clearance space are in steady state of continuously contacting; (3) The /d and screw coordinate deviation between ideal screw S actual screw S/r of the clearance characteristic element is caused by the movement of clearance screw S/ee . In local coordinate system o-xyz , the ideal screw coordinates of revolute and prismatic joints are S/dR = [0 0 1 0 0 0]T , S/dP = [0 0 0 0 0 1]T . The adjoint transformation caused by the clearance screw and clearance screw model[31] can be derived as

S/r = Ad(exp( S/ee )) S/d , ìï é ri - rd ïï ïïeRi = arccos( Si 3 ); S/eRi = êê ï ë di í ïï é ï = arccos( S ); S/ = ê ri - rd ï 6 ePi i ePi ï ê d ï ë i ï î

T rd ´ ri ri - rd ùú + , di  eRi úû T rd ´ ri ri - rd ùú , + di ePi úû

(30)

(31)

S P5

T

S P 6 ] , (34)

ìï ïïS P 4 = x2 - x1 , S P 5 = y2 - y1 , S P 6 = LP , ï SP SP SP í ïï ïïS P = ( x2 - x1 ) 2 + ( y2 - y1 ) 2 + LP 2 , î

(35)

where LP is the fitting length, x1 , x2 ( y1 , y2 ) denote the x(y) coordinates of the contact points. They are assigned to be random variables within x Î [- a ,  a ] , y Î [- b ,  b ] . And  a , b are the clearances in x and y directions, respectively. Substituting Eqs. (32) and (34) into Eq. (31), the expressions in local coordinate system are as follows: ì ï æL ö ï ï  eR = arccos ççç R ÷÷÷ , ï ï çè S R ÷ø ï ï ï ï é ï c 1 + c  2 ïíS/ = ê eR ê ï ï ëê 2[1 + c( 1 -  2 )] ï ï ï ïï  (c  + c  2 ) ï 0 a 1 ï ï 2 eR ï î

s 1 + s  2

(36)

2[1 + c( 1 -  2 )] T

 a (s  1 + s  2 ) ùú 0 , ú 2 eR û

CHINESE JOURNAL OF MECHANICAL ENGINEERING ìï æ LP ö÷ ïï ç ïï eP = arccos ççç ÷÷÷ , è SP ø ïï ïï é x1 + x2 y1 + y2 ïïS/ = ê í eR ê 2 2 ïï ê ( x1 + x2 ) + ( y1 + y2 ) ( x1 + x2 )2 + ( y1 + y2 ) 2 ë ïï ïï ùT ïï x + x2 y1 + y2 0 1 0ú . ïï ú 2 eP 2 eP û ïî (37) Mathematics model of manufacturing tolerances for the structural parameters Considering two kinds of parameters with manufacturing errors in the part, one is the link parameter and the other is the integrate parameter shown in Fig. 4(b). To the link parameters ci ( i = 0,1, , 6 ), they can be represented by S/vi when treated as virtual prismatic joints[23]. The kinematic variable is the tolerance ci as shown in Fig. 7(a). In machine coordinate system, screw coordinates of S/vi are

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machine is shown in Table 2, and the error models can be obtained from the output errors of driving system and transmission chain.

4.3

T é x - x Ai yBi - y Ai z Bi - z Ai ùú T S/vi = [ 0 v ] = ê 0 0 0 Bi , ê ú ci ci ci ë û (38)

where xi , yi , zi denote nominal coordinates of endpoints, ci is the nominal structural parameter of the link i . To the integrate parameter , the error model[22] of spherical link shown in Fig. 7(b) is used to analyze the effect of manufacturing and assembly tolerances. In local coordinate system o-xyz , ideal pose o ¢x ¢y ¢z ¢ and actual pose o* x* y* z * of the arc curved link is

Too¢

= Rot( x,  ),

Table 2. Number Outputs of drive motor group (rad) Transmission pair type Inputs (rad, mm)

(40)

where  ,  are angular tolerances of the arc curved link.  x,  z are the linear tolerances. Simplification is made when considered the actual conditions: only the angular tolerance  and the linear tolerance  s are taken into account; the others are ignored, as shown in Fig. 7(c). Then, pose of the arc curved link in the simplified local coordinate system are as follows:

Too¢ = Rot( y,  ),

(41)

Too* = Rot( y,  )Rot( y,  )Trans( z ,  s ).

(42)

4.4 Mathematics model of the input variable for active joints The transmission relationship of active joints of the

Description of manufacturing tolerances for structural parameters

Transmission relationship of input kinematic variable of active joints Joint 1

Joint 2

Joint 3

Joint 4

Joint 5

10

 20

 30

 40

 50

10

 20

 30

 40

 50

Gear pair

Ball screw 1

Gear-rack pair

Ball screw 2



1

d2

d3

d4

6

 1

d 2

d 3

d 4

 6

4.4.1 Angular input and angular output (1) Ideal kinematic equation of input-output relationship of the active joint 1 is

1 = 10

(39)

Too* = Rot( x,  )Rot( z ,  )Trans( x,  x)´ Trans( z,  z )Rot( x,  ),

Fig. 7.

z1 , z1 + z2

(43)

where z1 and z2 are the tooth number of pinion and big gear ring respectively. When the instantaneous meshing error is taken into account, the error model is derived as 1 = 10

F1 = ka

z1 + F1 , z1 + z2

(44)

6.875 F11¢ 2 + F12¢ 2 + TD112 + Td 112 + TD12 2 + Td 12 2 , mz2 (45)

where F1 is the angular boundary value of the one-way transmission error in an entire cycle of gear pair, and the parameters are shown in JIN[32] and HU, et al[33]. (2) The active joint 6 is directly driven by the driving motor unit 5, its ideal kinematic equation of input-output relationship and error model are

6 = 50 , 6 = 50 .

(46)

HU Shenghai, et al: Design and Accuracy Analysis of a Metamorphic CNC Flame Cutting Machine for Ship Manufacturing

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4.4.2 Angular input and linear output (1) Ideal kinematic equation of input-output relationship and error equation that considered the effect of accumulated error and variation quantity in ball screw of active joint d2 are

d2 =

 20

p1 , d 2 =

2π

 20 p1 + p1 , 2π

(47)

where p1 is the lead of ball screw 1, p1 is the boundary value of location error. To the active joint d4 , its input-output relationship and error expression can be obtained when combined the physical parameters of ball screw 2, and yields

d4 =

 40

p2 , d 4 =

2

 40 p2 + p2 . 2π

 30 2

mz , d 3 =

Ft 3 = 6.875ka

 30 mz + Ft 3 , 2

F31¢ 2 + F32¢ 2 + TD 312 + Td 312 ,

-1  g st1 ( ) = g st1 ( )( g streal 1 ( )) .

(49)

(50)

where m and z is the modulus and tooth number of the gear in gear-rack pair, Ft 3 is the linear boundary value of the one-way transmission error[32, 33]. Kinematic error model of overall configurations during a working cycle An important assumption is made before deriving the kinematic error models: the input kinematic variable error of the locked joint is zero, but the clearance screw is still active. When the combined effects of the error sources are taken into account, the actual forward kinematics equation of straight hole cutting configuration is

/ˆ /ˆ /ˆ gstreal 2 ( ) = exp( c0 Sv 0 ) exp( c1 Sv1 ) exp(( d 2 + d 2 ) S sr 2 ) ´ exp((d + d ) S/ˆ ) exp(c S/ˆ ) exp(c S/ˆ )´ 3

 ) = exp(c0 S/ˆv 0 ) exp((1 + 1 ) S/ˆsr1 ) exp(c1S/ˆv1 )´ exp((d 2 + d 2 ) S/ˆsr 2 ) exp((d3 + d3 ) S/ˆsr 3 ) ´ exp(c S/ˆ ) exp(c S/ˆ ) exp(c S/ˆ ) g real (0), (51) 2 v2

3 v3

4 v4

st1

S/srj = Ad(T jS )S/rj ,

(52)

where S/vi can be obtained from Eq. (38), S/srj ( j = 1, 2,3) is the representation in machine coordinate system of actual joint screw and S/rj is established from Eq. (32) to Eq. (35). T jS is the transformation matrix, gstreal 1 (0) denotes the initial configuration and is /ˆ /ˆ gstreal 1 (0) = exp( e 6 S e 6 ) exp( e 7 Se 7 ) g st1 (0) ´ Rot( y,  )Trans( z ,  s ),

(53)

3

sr 3

2 v2

3 v3

exp(( 6 +  6 ) S/ˆsr 6 ) exp(c4 S/ˆv 4 ) exp(( 7 +  7 ) S/ˆsr 7 ) gstreal 2 (0), (55)

/ˆ gstreal 2 (0) = exp( e1 Se1 ) gst 2 (0)Rot( y,  )Trans( z ,  s ),

(56)

/e1 is screw coordinates of locked joint 1, and the where S /e6 , S/e7 . derivation process and expression are similar with S Then, the error model can be derived through replacing the subscript 1 to 2 in Eq. (54), which is equivalent to substituting the expression in Eq. (25) and Eq. (55). The actual forward kinematics equations of bevel cutting configuration are as follows:

4.5

gstreal 1 (

(54)

Therefore, the position and orientation errors of current configuration can be obtained through every component in  gst1 ( ) that respects to machine coordinate system. Similarly, the actual forward kinematics equations combined with the error sources expressions in pose adjusting configuration are derived as

(48)

(2) The input-output relationship and error model of the active joint d3 are as follows: d3 =

where S/e6 , S/e7 are screw coordinates of locked joints in machine coordinate system, and are obtained by Eq. (52) /eR 6 , S/eR 7 should be established through Eq. (36). while S  ,  s are structural tolerances of arc curved link. Combining Eq. (51) and Eq. (19), the error model of straight hole cutting configuration is

/ˆ /ˆ /ˆ gstreal 3 ( ) = exp( c0 Sv 0 ) exp(( 1 + 1 ) S sr1 ) exp( c1 Sv1 ) ´ exp((d 2 + d 2 ) S/ˆsr 2 ) exp(( d3 + d3 ) S/ˆsr 3 ) ´ exp(c2 S/ˆv 2 ) exp(c3 S/ˆv 3 ) exp(c4 S/ˆv 4 ) ´ exp(( +  ) S/ˆ ) g real (0), 7

7

sr 7

st 3

/ˆ gstreal 3 (0) = exp( e 6 Se 6 ) g st 3 (0)Rot( y,  )Trans( z ,  s ).

(57) (58)

The error model of bevel cutting configuration is also derived by replacing the subscript 1 to 3 in Eq. (56), which means to substitute expression in Eq. (26) and Eq. (57). So far, the kinematic error models of the machine are completely established, which means that the mapping relationship between kinematic errors of end-effector and the original parameters of the machine (including fitting, manufacturing and assembly tolerances), errors in the control system and transmission chains of active joints have already been derived.

5

Results of Numerical Simulations and Experimental Studies

5.1 Numerical simulations and processing experiment of the Metamorphic CNC flame cutting machine In this part, a large size eccentric circular hole with upper

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and lower bevels on the cylindrical surface is chosen as the basic pass. For each component of the basic pass, parameters are reported in Table 3, and for each component of the mechanism, nominal dimensions are reported in Table 4. Table 3.

Data relative to the basis pass and cutting tool shown in Fig. 2

Pass parameters (mm) R 5750 RS 2 5777

S 60 e 1650

Table 4. Manufacturing variables Nominal dimensions Tolerances Manufacturing variables Nominal dimensions Tolerances

RS 1 5783 r 1350

Bevel angle (°)

Cutting tool parameters (mm)

e

f

r0

s0

L

[62, 30]

[30, 62]

4

15

200

Data relative to the kinematics shown in Fig. 4 and Fig. 7 c0 /mm

c1 /mm

c2 /mm

c3 /mm

c4 /mm

1365

65

320

546.5

531.5

1

0.1

0.2

0.4

0.4

c5 /mm

c6 /mm

s0 /mm

 /(°)

440

300

15

44

0.3

0.2

0.1

0.1

Fig. 8 shows the basis description of the processing task, and Fig. 9 shows the results of inverse solutions according to parameters of the basic pass. Fig. 10 shows the inverse solutions of active joints in straight hole cutting configuration and bevel cutting configuration while the pose adjusting configuration is not included.

Fig. 8.

Basis description of processing task

Fig. 9.

Fig. 10.

Inverse solutions of the cutting tool

Inverse solutions of active joints of the machine

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HU Shenghai, et al: Design and Accuracy Analysis of a Metamorphic CNC Flame Cutting Machine for Ship Manufacturing

Fig. 11 shows the processing experiment of the CNC flame cutting machine. As shown in case (a), the machine is installed on the cylindrical surface by the adjustable frame, and the processing begins when leveling and basic function test have been done. Case (b) shows the intermediate process of the experiment, and it can be drawn that the operation of the machine is stable and the velocity is suitable for the cutting thickness through the excrete direction of the spark. Fig. 12 shows the cutting effect after the processing experiment, while case (a) shows the surface of straight hole contour, and case (b) shows the surface of bevel contour that contains upper, lower bevels and bevel root-cutting.

5.2 Numerical simulation of kinematic error models and trajectory tracking experiment Table 5 shows the clearance parameters of revolute joint and prismatic joint that correspond to Fig. 6. For each component of the mechanism, tolerances are also reported in Table 4. And for each input variable of active joints, error boundary values of the control system and transmission chains are reported in Table 6. Then, the numerical simulations are carried out based on interval analysis method[20, 34]. Note that, the results are also relative to the workpiece coordinate system rather than machine coordinate system. Table 5.

Clearance parameters shown in Fig. 6

Joint number Joint 1

0.076

60

(mm)

Joint 6 and 7

0.029

14

(mm)

Table 6.

Processing experiment of the cutting machine

Fig. 12. Local amplification and micro-show of the cutting effect after processing experiment

LR

Revolute joint Prismatic joint

Fig. 11.

a

Joint 2 and 3



a

b

LP

 0.05

0.05

40

Error boundary values of the input variable of active joints

Error variables

 i 0 (i = 1, , 5)

p j ( j = 1, 2)

(°)

(mm)

Tolerances

0.5

0.1 / 300

F 1 (°)

Ft 3 (mm)

0.026

0.14

The trajectory tracking experiment completed by a laser tracker (AT901-LR) is divided into two groups with the machine is located at the ground and the workpiece surface respectively. What should be mentioned are that the orientation results are obtained from converting position results of endpoints in the centerline of cutting tool and results are all converted to workpiece coordinate system. Fig. 13 shows the results of kinematic error models and tracking experiment of straight hole cutting configuration. Fig. 14 shows the change law of error radius calculated by original data rather than extreme value. Fig. 15 shows the results of kinematic error models of the lower and upper bevel cutting configuration. Fig. 16 shows two kinds of error space of the initial point. Fig. 17 shows the endpoint’s trajectory of cutting tool in pose adjusting configuration.

Fig. 13. Results of kinematic error models and tracking experiment of straight hole cutting configuration

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Fig. 10. However, the machine workspace is not given to reflect the processing range in this paper, while the movable intervals of active joints are different when the machine is installed on different surface, and it is impossible to traversal all installation conditions.

Fig. 14. Change law of error radius of four mapping parameters

Fig. 15.

Results of kinematic error models of bevel cutting configuration

Fig. 17. Endpoint’s trajectory of cutting tool in pose adjusting configuration

Fig. 16. Error space of the initial point in lower bevel cutting configuration

5.3

Analysis of the resulted data receiving from simulations and experiments As shown in Fig. 8 to Fig. 12, these results prove that the designed machine can satisfy the requirements of variable trajectory and variable pose to cut continuously and automatically on complex spatial curved surface. Other kinds of holes such as orthogonal hole, eccentric elliptical hole et al on complex spatial curved surface of hull can be processed by using the same procedures shown in Fig. 8 to

The results shown in Fig. 13 to Fig. 17 prove that the kinematic error models established by three types of error sources from the original designing parameters are effective and credible, although the concrete parameters to extreme value are not given. These results also show that the characteristic of multiple configurations and variable constrains must be considered for accuracy analysis of metamorphic mechanism. It can be also drawn that different results would be obtained in case of different configuration with the same parameters, which means it is difficult to obtain the optimal results for precision design by inversing the error model of each independent configuration. The differences shown in Fig. 15 indicate that kinematic error bound would change at the time of configuration transformation, and the reason is the change of configuration constraints. Fig. 14 shows that calculated errors are not constant and the obtained bounds are related to the extreme values, and those are caused by error sources and current configuration. The comparative results shown in Fig. 16 and the non-zero results in Y direction shown in Fig. 17 illustrate that considering the effect of joint clearances is indispensable for accuracy analysis, otherwise, the numerical results of accuracy analysis and the subsequent accuracy design and error compensation would be not in accord with the actual movement of mechanism or machine.

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HU Shenghai, et al: Design and Accuracy Analysis of a Metamorphic CNC Flame Cutting Machine for Ship Manufacturing

The kinematic accuracy of the designed machine is obtained from the above results, but is not equivalent to the machining accuracy, while dynamic factors such as thermal deformation and component deformation et al are not considered in this paper.

6 Conclusions (1) This paper carries out the variable trajectory and variable pose characteristics of non-contact cutting tool for the continuous and automatic requirements of cutting large size fabrication holes on complex spatial curved surface. It is useful for the subsequent researches to develop kinds of cutting machine such as laser, plasma cutting machine, and so on. (2) A metamorphic CNC flame cutting machine that can change its topological structures is developed to meet the kinematic characteristics of cutting tool. The processing experiment and simulation results show that metamorphic principle can adapt to variable working tasks, and also indicate the processing capacity and effect of the machine used on spatial curved surface of hull. (3) Kinematic error bounds of the designed machine are obtained through numerical solutions of error models established by the original tolerance parameters, and are verified by trajectory tracking experiments during a working cycle. The results illustrate that considering the effect of joint clearances is indispensable and the characteristics of multiple configurations and variable constrains of metamorphic mechanism have to be taken into account in accuracy analysis. It can also be drawn that the machine motion is waving around the theoretical value within a ‘large’ range and error compensation is necessary and underway. (4) The designed machine, which has been used in Bohai shipbuilding heavy industry co ltd after error compensation, has solved the key problem in ship manufacturing. Furthermore, it can also be extended to pressure vessel and large pipe manufacturing industry and so on. References [1] LIANG Guifang. Cutting technology handbook[M]. Beijing: China Machine Press, 1997. (in Chinese) [2] National Natural Science Fund Committee and Materials Science Branch. Report the mechanical engineering discipline development strategy (2011-2020)[M]. Beijing: Science Press, 2010. (in Chinese) [3] YANG Wenyu, YIN Zhouping, SUN Ronglei. Fundamentals of digital manufacturing[M]. Beijing: Beijing Institute of Technology Press, 2005. (in Chinese) [4] KOMATSUBARA H., MITOME K I, SASAKI Y A. A new cutting machine for elliptical cylinder[J]. Transactions of the Japan Society of Mechanical Engineers: Part C, 2007, 19(3): 891–896. [5] GORDON S, HILLERY M T. Development of a high-speed CNC cutting machine using linear motors[J]. Journal of Materials Processing Technology, 2005, 166(3): 321–329. [6] WANG Guodong, YAN Xiang’an, XIAO Juliang. Study on NC cutting of welding groove for intersecting of pipe and annulus[J]. Chinese Mechanical Engineering, 2005, 16(6): 561–563. (in Chinese)

[7] SUN Yingda, CHEN Qiong. Control of a pipe-cutting machine[C] //International Conference on Measuring Technology and Mechatronics Automation, Changsha, China, March 13–14, 2010: 845–848. [8] KLAUS D. Modern cutting machines and techniques in shipbuilding industry[J]. Svetsaren (a welding review published by ESAB), 2002, 57(1): 22–26. [9] TAYLOR R. Flame-cutting apparatus: European, EP 1867425. A1[P]. 2007-06-12. [10] RAIMONDI S. Laser cutting machine for metal sheets: European, EP2060358. A1[P]. 2007-11-14. [11] MARUYAMA Y, KAWAGUCHI T. Torch angle setting apparatus: USA, US6201207. B1[P]. 2000-03- 29. [12] WANG Zongyi, HU Shenghai, ZHAO Shijun. Design of big intersecting circle flame cutting machine[J]. Journal of Harbin Engineering University, 2003, 24(3): 258–262. (In Chinese) [13] DAI Jiansheng, ZHANG. Qixian. Metamorphic mechanisms and their configuration models[J]. Chinese Journal of Mechanical Engineering, 2000, 13(3): 212–218. [14] KUO Chinhsing, DAI Jiansheng, YAN Hongsen. Reconfiguration principles and strategies for reconfigurable mechanisms[C]//ASME/ IFToMM International Conference on Reconfigurable Mechanisms and Robots, London, United Kingdom, June 22–24, 2009: 1–7. [15] LEE K I, YANG S H. Measurement and verification of position-independent geometric errors of a five-axis machine tool using a double ball-bar[J]. International Journal of Machine Tools and Manufacture, 2013, 70: 45–52. [16] SEBASTIAN O H Madgwick, ANDREW J L Harrison, PAUL M. S, et al. Measuring motion with kinematically redundant accelerometer arrays: theory, simulation and implementation[J]. Mechatronics, 2013, 23(5): 518–529. [17] TSAI M J, LAI T H. Accuracy analysis of a multi-loop linkage with joint clearances[J]. Mechanism and Machine Theory, 2008, 43: 1141–1157. [18] FRISOLI A, SOLAZZI M, PELLEGRINETTI D, et al. A new screw theory method for the estimation of position accuracy in spatial parallel manipulators with revolute joint clearances[J]. Mechanism and Machine Theory, 2011, 46(12): 1929–1949. [19] HUANG Yonggang, DU Li, HUANG Maolin. Screw theory based error modeling method of robot mechanisms[J]. Journal of Harbin Institute of Technology, 2010, 42(3): 484–489. (In Chinese) [20] WU Weidong, RAO S S. Interval approach for the modeling of tolerances and clearances in mechanism analysis[J]. Journal of Mechanical Design, 2004, 126(4): 581–592. [21] WU Weidong, RAO S S. Uncertainty analysis and allocation of joint tolerances in robot manipulators based on interval analysis[J]. Reliability Engineering and System Safety, 2007, 92: 54–64. [22] CHAKER A, MLIKA A, LARIBI M A, et al. Clearance and manufacturing errors’ effects on the accuracy of the 3-RCC spherical parallel manipulator[J]. European Journal of Mechanics A/Solids, 2013, 37: 86–95. [23] KUMARASWAMY U, SHUNMUGAM M S, SUJATHA S. A unified framework for tolerance analysis of planar and spatial mechanisms using screw theory[J]. Mechanism and Machine Theory, 2013, 69: 168–184. [24] YU A, BONEV I A, ZSOMBOR-MURRAY P. Geometric approach to the accuracy analysis of a class of 3-DOF planar parallel robots[J]. Mechanism and Machine Theory, 2008, 43(3): 364–375. [25] BRIOT S, BONEV I A. Accuracy analysis of 3T1R fully-parallel robots[J]. Mechanism and Machine Theory, 2010, 45(5): 695–706. [26] DING Han, ZHU Limin. Geometric theories and methods for digital manufacturing of complex surface[M]. Beijing: Science Press, 2011. (in Chinese) [27] REN Fei, SUN Yuwen, GUO Dongming. Combined reparameterization-based spiral toolPath generation for five-axis sculptured surface machining[J]. International Journal of Advanced Manufacturing Technology, 2009, 40: 760–768.

CHINESE JOURNAL OF MECHANICAL ENGINEERING [28] YAO Yanan, ZHENG Ce, YAN Hongsen. Motion control of cam mechanisms[J]. Mechanism and Machine Theory, 2000, 35: 593–607. [29] WANG Delun, DAI Jiansheng. Theoretical foundation of metamorphic mechanism and its synthesis[J]. Chinese Journal of Mechanical Engineering, 2007, 43(8): 32–42. (in Chinese) [30] GRIGORE G. Mobility of mechanisms: a critical review[J]. Mechanism and Machine Theory, 2005, 40: 1068–1097. [31] MURRAY R M, LI Zexiang, SASTRY S S. A mathematical introduction to robotic manipulation[M]. Florida: CRC Press, 1994. [32] JIN Taiyi. Theory and application of precision[M]. Hefei: Press of University of Science and Technology of China, 2005. (In Chinese) [33] HU Shenghai, SHI Jianbin, ZHAN Guochen, et al. Accuracy analysis for high precision space curve numerical control cutting machine[J]. Journal of Harbin Engineering University, 2001, 22(6): 97–100. (in Chinese) [34] KIM H S, YONG J C. The kinematic error bound analysis of the stewart platform[J]. Journal of Robotic Systems, 2000, 17(1): 63–73.

Biographical notes HU Shenghai, born in 1954, is currently a professor at Harbin Engineering University, China. His research interests include naval gun weapon system, NC machining technology, mechanical

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design theory. Tel: +86-451-82519055; E-mail: [email protected] ZHANG Manhui, born in 1991, is currently a PhD candidate at College of Mechanical and Electrical Engineering, Harbin Engineering University, China. His main research interests include metamorphic mechanism, NC machining technology. Tel: +86-451-82519055; E-mail: [email protected] ZHANG Baoping, born in 1990, is currently a master candidate at College of Mechanical and Electrical Engineering, Harbin Engineering University, China. E-mail: [email protected] CHEN Xi, born in 1991, is currently a master candidate at College of Mechanical and Electrical Engineering, Harbin Engineering University, China. E-mail: [email protected] YU Wei, born in 1989, is currently a master candidate at College of Mechanical and Electrical Engineering, Harbin Engineering University, China. E-mail: [email protected]

Appendix Appendix A: Subassembly drawing of metamorphic CNC flame cutting machine

Fig. 18.

Virtual prototype of components of the designed CNC flame cutting machine

Appendix B: Derivation of expression of passive joint in closed loop

To convert the input variables 7 , 7 of passive joint that appeared in the error models to active joints in closed loop, the structural equation can be spread as

(59)

Assume that  = tan( 7 / 2), because of the structural limits in closed loop structure, then  7 Î [-π / 2, π / 2] and  Î [- 1, 1]. By substituting triangle substitution equation to Eq. (59) and yields ì ï 2 1-  ï sin  7 = , cos 7 = , ï 2 ï 1+  1+  2 ï ï í ï ï N  N 2 -T 2 + M 2 ï = , ï ï T +M ï î

(61)

Therefore, the explicit expression of 7 is derived as

c42 + c6 2 + d 42 - c52 + 2d4 c4 sin 6 = (2c4 c6 + 2d 4 c6 sin 6 ) cos 7 + 2d 4 c6 cos 6 sin 7 .

ìïT = c 2 + c 2 + d 2 - c 2 + 2d c sin  , 4 6 4 5 4 4 6 ïí ïïM = 2c4 c6 + 2d 4 c6 sin 6 , N = 2d 4 c6 cos  6 . î

2

(60)

7 = 2arctan  .

(62)

By differentiating Eq. (62) with respect to time and yields

7 =

2 . 1+  2

(63)

Because of the kinematic variables of active joints changing in different configuration, Eq. (60) and Eq. (63) should also change correspondingly.