Integrated Design of Robotic Welding Systems

Copyright © IF AC Infonnation Control Problems in Manufacturing, Vienna, Austria, 2001

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INTEGRA TED DESIGN OF ROBOTIC WELDING SYSTEMS i

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A. Pashkevichl.3, A. Dolgui ,0. Zaikin , K. Semkin

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i/ndustrial Systems Optimisation Laboratory. University of Technology of Troyes 12, rue Marie Curie B.P. 2060. Troyes, 10000, France , -Department of Computer Networks and Information Community. Technical University of Szczecin. 49. Zolnierska. 71-210, Szczecin. Poland 3Robotic Laboratory, Belarusian State University, of Informatics and Radioelectronics, 6. P.Brovka St. , Minsk. 220600. Belarus

Abstract: The paper focuses on the integrated design of a redundant two-manipulator robotic system taking into account particularities of the arc welding technology. It have been proposed a novel formulation and closed-form solution of the inverse kinematic problem that deals with explicit definition of the weld joint orientation relative to the gravity. There have been carried out detailed investigation of singularities and uniqueness-existence topics. The presented results are implemented in a commercial software package and verified for real-life applications. Copyright © 2001 IFAC Keywords : redundant manipulators, inverse kinematic problem, positioning systems.

along the weld joint with the prescribed velocity and orientation relative to the joint, while the weld must be also proper oriented relative to the gravity vector. For this reason, typical welding station (Fig. 1) includes several computer-controlled machines: a 6axis industrial robot (tool manipulator); a 1-2 axis positioner (object manipulator); and, optional, a 2-3 axis gantry (robot manipulator). Such arrangement forms redundant kinematic system, which does not possess closed-form solution for optimal configuration of all machinery. Because of complexity of this problem, it is usually decomposed in several separate tasks, which are solved sequentially: I) optimising weld joint orientation relative to the gravity vector; 2) optimising the welding tool orientation relative to the weld joint; and 3) optirnising robot base location . Each of these steps requires multiple coordinate transformation for corresponding machines (both direct and inverse). However, in robotic literature, the main research activity focuses on 6 d.o.f. or redundant robots, which are investigated in details. To our knowledge, only G.Bolmsjo (1987) and G.N icoleris (1990) from the Lund Institute of Tech-

I. INTRODUCTION Welding technology is a traditional application area of industrial robots that encourages intensive development of computer-aided design and programming tools. However, there are still a number of theoretical problems, which at present stage are overcome in industry by combining simulation with expertise of a designer. This paper concentrates on one of these problems: the inverse kinematics of the welding positioner. Such emphasis to kinematics is originated from the dual nature of robotic control, which requires defining both logical structure of the manufacturing task and specifying some spatial relations. Obtaining these spatial relations is very tedious and time-consuming process, which is typically from 10 to 100 times longer then the welding circle (Bolmsjo, 1999). In arc welding applications, kinematic capabilities of a 6 d.o.f. robot are not usually sufficient to ensure required working envelope and/or desired orientation of a welding torch. The robot must move the tool

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For the linear joints, a moving frame with specific definition of axes can describe the weld geometry. In this paper, this frame is defined so that the X-axis is directed along the weld joint; the Y-axis specifies the weld torch approaching direction (and normal to the weld joint); and the Z-axis completes the right hand oriented frame and thus shows the weaving direction. So, the linear weld kinematic model can be described by homogenous parametric equation

nology (Sweden) investigated positioner kinematics for welding applications. And even in sophisticated industrial CAD packages, such as ROBCAD (Tecnomatix Technologies) and IGRIP (Deneb Robotics), the optimisation process is still semi-automatic and employs direct/inverse kinematics for robots, but only direct kinematics for positioners. This paper expands results of G.Bolmsjo and G.Nicoleris by proposing closed-form solutions for the optimal weld joint orientation and careful investigation of the solution singularities and existence issues. There are also proposed expressions for positioner configuration indices, which ensure a unique solution of the problem for a single input and continuous solutions for the sequence of inputs defining the welding path. The latter is extremely important for coordinated control of the positioner and the robot in the Cartesian space, when change of configuration is not allowed.

where I is the welding torch displacement, L is the weld length, and Wo is the 4x4 homogenous matrix corresponding to the weld starting point. For the cirClllar joints, a similar approach is used, but the moving frame is computed to ensure tangency of the weld path and the X-axis at every point. It is evident that the initial frame Wo is subject to the rotational transformation and the weld kinematic model is described by the following parametric equation

The reminder of this paper is organised as follows . Section 2 describes the main notations and the kinematic description of the welds. Section 3 is devoted to formal statement of the weld joint orienting problems. In Section 4, direct kinematic model of a general welding positioner is derived. Section 5 gives the inverse kinematic solution as well as detailed investigation of singularities and uniquenessexistence issues. In Section 6, industrial implementation is presented and real-life industrial applications are described. And, finally, Section 7 summarises the main contributions of the paper.

where Pe is the position vector of the centre of the circle, RMo is the 3x3 rotational part of the homogenous matrix W(O), Re (v) is the general rotation matrix around the axis e, and the rotation angle v=llr is computed using the welding torch displacement I and the circle radius r.

2. KINEMATIC DESCRIPTION OF THE WELDS Hence, the world location of the weld joint frame is described by the product of the homogenous matrices

Spatial location of the welding object, as a general rigid body, can be defined by a single frame that incorporates 6 independent parameters (3 Cartesian coordinates and 3 Euler angles). However, defining geometry of each weld requires some additional efforts, depending on the joint profile. Since capabilities of modem commercial robotic systems allow to process two types of contours (linear and circular), only these cases are considered below.

where the left superscript "0" refers to the world coordinate system, the matrix °TPB defines absolute (world) location of the positioner base, the matrix PF TwB describes the workpiece base location relative to the positioner mounting flange, and the matrix function P(q) is the positioner direct kinematic model with the axis coordinate vector q. To ensure the product quality and to increase welding speed, the weld joint should be proper oriented relative to the vector of gravity. The exact interrelations between these parameters are not sufficiently well known and require empirical study in each particular case. But practising engineers have developed rather simple rule of thumb, that is widely used for both on-line and off-line programming: "the weld should be oriented in the horizontal plane so that the . . . ·elding torch is vertical, if possible" (Bolmsjo, 1987). It is obvious, that simulation-based approach requires numerical measures of the "horizontally" and the "verticality", which are proposed below.

Fig. I. Welding robotic station

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It should be stressed that both formulations require two independent input parameters (two angels or a unit vector), however they differ by the elements of the matrix °WR they deal with (the third row or the second column). As a result, the second formulation is less reasonable from technological point of view.

Let us assume that the Zo-axis of the world coordinate system is strictly vertical, and, consequently, the XoYo-plane is horizontal. Then, the weld orientation relative to the gravity vector can be completely defined by the angles: ~

°a Z w (4) = atan2-o z s ...

4. DIRECT KINEMATICS Because the weld orientation relative to the gravity is completely defined by two independent parameters, a universal welding positioner has 2 joint axes. Its kinematic model includes four linear parameters (a" d" a2, d2) and one angular parameter a that defines direction of the Axis, . Without loss of generality, the Axis2 is assumed to be normal to the faceplate and directed vertically while q,=O. So, the considered system may be described by the equation:

where ge [-1tI2; 1tI2] is the weld slope (or pitch), i.e. the angle between the welding direction n", and the XoYo plane; and ~e ]-7t; 7t] is the weld roll, i.e. the angle between the approaching direction Sw and the vertical plane that is parallel to the vector n", and Zoo It should be noted that it is possible to introduce alternative definition of the weld roll, as the angle between the approaching direction Sw and the vertical axis Zo:

~' = atan2

kr + kr Is!.

where

(5)

PB TI = Tx{al)'Tz {d l )· Ry{-a) 'T2 = Ry{a)·Tx {a2) ' Tz {d 2 )

In can be proved that the interrelation between ~ and ~' is given by the equation cos(9)·cos(~)=cos(~')

The scalar expressions for the non-trivial components of the matrix P(q"q2) are the following:

and both (9, ~) and (9, ~') can be used consistently.

3. ORIENTING OF THE WELD JOINT

nx = (cl

+C~ VI )C2 -Sa S,S2

nv =SaSIC2 +CIS 2 ;

In robotic welding station, the desired orientation of the weld relative to the gravity is achieved by means of a positioner, which adjusts the slope and the roll angles (9,~) by alternating the axis coordinates q. Using the definitions from the previous section, the considered problems can be stated as follows.

(8)

n z = C a S a VIC 2 + Ca S IS 2 ; Sx

= -(Cl + C~ VI )S2 -SaSIC2 ;

sI' =-SaSIS 2 +CIC 2 ; SZ

Direct Problem. For given values of the positioner axis coordinates q, as well as known transformation matrices °TpB , PFTWB and W, find the weld frame location relative to the world °w and the slopelroll orientation angles (9, ~).

(9)

=-Ca S a VIS 2 +Ca S IC 2 ;

Px = (Cl + C~ VJ a 2 + CaSaVI . d 2 + Q,

Py = Sa SI 'a2 - CaS} . d 2

Inverse Problem 1. For given values of the slope/roll orientation angles (9, ~), as well as known transformation matrices °TpB ,PFTwB and W, find the values of the positioner axis coordinates q.

p= = CaSaV} 'a2 +(C}

(I 1)

+S~ VJd 2 +d}

where the vectors n, s, a, p define the upper 3x4 block, and C. S, V denote respectively cos(.), sin(.), vers(.) of the angle specified by a subscript.

There exist also another version of the positioner inverse problem (Nikoleris, 1990) that deals with a reduced version of the basic kinematic equation (3):

5. INVERSE KINEMATICS In accordance with the Section 3, solving the inverse kinematic problem for the positioner means finding the axis angles (q" q2) that ensure desired world orientation of the weld joint, which is defmed by a pair of orientation angles (Problem I) or a unit vector (Problem 2). Let us consider these cases separately.

Inverse Problem 2. For given values of the approach vector os"' as well as for known transformation matrices °TPB , PF TwB and the normal vector orientation relative to the object base sw, find the values of the positioner axis coordinates q.

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5.1. Solution of the Inverse Problem 1

Further expansion of P(q) and regrouping leads to the following scalar equations:

Since the weld joint orientation angles (e,~) or (e, ~/) completely define the third row of the orthogonal 3x3 matrix °WR , the basic kinematic equation (3) can be rewritten as

(I-S~VI ~x+SaSI uy+SaCaVI Uz = C 2 w x - S 2w " -SaSlux +CllI y +CaSlu z =S2wx+C2Wy SaCa Vlu x -CaSlu\"

+(I-C~VI)lI=

(19)

= Wz

where the third one can be solved for ql where T{=[0 0 I] and a sUbscript 3x3 denotes the rotational part of the corresponding homogenous transformation matrix. Then, after appropriate matrix multiplications, it can be converted to the form (13)

and the first two equations yield q2: (21) Further substitution yields the following mutually dependent scalar equations C a S a VI C 2 + Ca SI S2 =

three

where Vx = lIx+Sa (SI ur-VIlIxz );

Vy = CIUy-SIUxz'

VI"

Ca SI C2 -Ca S a VI S 2 =vr

Therefore, the solution of the second inverse problem also gives two pares of the angels (qh q2).

(14)

CI+S~VI=Vz' 5.3 Existence and Singlllarities

The third of them can be solved for ql V

ql = ± acos z

_S2 2a

As follows from the Eqs. (15), (16) and (20), (21), the inverse kinematic problems possess a solution only for certain sets of input data that can be treated as the positioner "orientation workspace". For the first inverse problem, investigation of (15) shows that ql can be definitely computed if and only if

(15)

Ca while the first and the second ones yield q2: (16)

-cos(2a):5 v=:51.

(22)

Taking into account geometrical meaning of vz , this condition can be presented as follows:

Therefore, the solution of the first inverse problem gives two pares of the positioner axis angels (qh q2) .

Proposition la. For the inverse problem I, the angle if and only if the angle X between Z-axes of the conjugate frames [°WR ] T and PF T. [ T WB' W R ] IS less than or equal to (x-2a).

ql can be computed definitely,

5.2. Solution of the Inverse Problem 2

For the second formulation, the input data define the second column of the matrix °WR , so the basic kinematic equation (3) can be rewritten as

In the typical for industrial application case, when Zaxis of the workpiece frame is parallel to positioner Axis2, the expression for q2 can be rewritten as

T

where 11=[0 1 0]. Then, after appropriate matrix multiplications, this equation can be converted to

which detailed investigation yields:

Proposition lb. For the inverse problem 1, the angle q2 (for given qd can be computed uniqu~~v, and only if Z-axes of the conjugate frames [WR ] and (FTwB 'WR ] T are not coincide, i.e. X>O. Otherwise, if these axes coincide (i.e. X=O), then ql=O and any value of q2 satisfy the kinematic equation.

( 18)

if

where

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Therefore, for the first inverse problem, the singularity exists only with respect to the positioner Axis2, while it is oriented strictly vertically and upward (q,=O).

F or the first inverse problem, the configuration index is defined trivially (see Eq. (15», as the sign of the coordinate q, : (26)

However, for the second inverse problem, the singularity may also arise for the Axis, because atan2 is indefinite if 11<.:=0 and u.,=O (or if uz=wz ). To ensure definite computing of q]' -it is also required that acos argument belongs to [-1; 1]. After appropriate rearranging, this condition can be presented as

But for the second problem, such index must identify the sign of the second term only (see Eq. (20». So, it should be defined as

From geometrical point of view, the index M2 indicates relative location of two planes, passing the Axis,. The first of them is obtained by rotating of XoZo around the Axis, by the angle q,. And the second one is passed via the Axis, and the vector u.

and the final results may be summarised as follows :

Proposition 2a. For the inverse problem 2, the angle q, can be computed definitely, if and only if the angles )1, Tl between the positioner Axis" Axis2 and the vectors u, w respectively satisfy the inequalities { 1l+CJ.-1t/2~1l~

ll-a+1t/2

-1l-a+~2~1l~-Il+a+3~2

5.5 Optimal Orienting of the Weld Joint

As adopted by practising engineers, the optimal weld orientation is achieved when the approaching vector is strictly vertical and consecutively, the weld direction vector lies in the horizontal plane, i.e. (9, ~)=(O, 0) and °s",= [00 1). Let us investigate this particular case in details.

(25)

and, additionally, )1*0 and )1*1t. Otherwise, if ()1, Tl)=(O, 7tl2-a) or ()1, Tl)=(1t, 7tl2+a), any value of q, satisfy the kinematic equation.

For the both inverse problems, substation of (9,~) and os", in the inverse kinematic equations yield the similar result:

In accordance with (21), computing of q2 can fail only in the case of wx=w,,=O, i.e. for Tl=O or Tl=1t. Geometrically, it corresponds to the vector w, which is normal to the positioner faceplate. So, the existence and uniqueness of solutions for q2 are subject to the following proposition:

(28)

So, the existence condition is reduced to s~, ~ -C2a .

Proposition 2b. For the inverse problem 2, the angle q2 (jor given q,) can be computed uniquely, if and only if the angle Tl between the positioner Axis2 and the vectors w satisfy the conditions Tl*O and Tl*1t. Otherwise, ifTl=O or Tl=1t, any value of q2 satisfy the kinematic equation.

It means, that the "working space" of the positioner does not include the cone with the downward directed central axis and the aperture angle 4a. And, thereby, corresponding welds can not be optimally oriented. But it have been proved that applying the first inverse problem with ~' =max{O; 2a + Tl-1t}

the orientation of such welds can be essentially improved and approached to the optimal one.

Therefore, for the second inverse problem, the singularity may exist for both axes, when u is parallel to the Axis, or w is parallel to the Axis2•

6. INDUSTRIAL IMPLEMENT A nON 5.4. Positioner Configurations

The developed algorithms have been successfully implemented on the manufacturing floor, in ROBOMAX CAD package (Buran Co, RussiaUSA), which is a powerful integrated system for computer-aided design and programming of welding robotic cells. In contrast to other robotic CAD systems, ROBOMAX incorporates a number of numerical routines which enable user to design welding robotic cells (lines) in semi-automatic mode. And all these tools require fast and reliable coordinate transformation for the welding positioner.

Similar to other manipulating systems, the positioner inverse kinematics is non-unique because of existence of two solution branches. However, both offline programming and real-time control require distinguishing the solutions to ensure continuity of the positioner motions. For this reason, the direct kinematics must yield additional output, configuration index M=± 1 describing positioner posture, which is also used as additional input for the inverse transformation, to produce a unique result.

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7. CONCLUSION The developed kinematic models allow to coordinate movements of two manipulators (the robot and the positioner) taking into account particularities of the welding technology. By using this technique together with the workpiece CAD-data, it is also possible to achieve essential time reduction of the design and programming of the robotic welding station. Particular contribution of this paper deals with the inverse kinematics of the 2-axis positioner, which is a key issue in the coordinated control of the welding robotic system. It have been proposed a novel formulation and closed-form solution of the inverse kinematic problem that deals with explicit definition of the weld joint orientation relative to the gravity.

Fig. 2. Contour plot of the function ~/(q"q2)'

The obtained results have been implemented in commercial software that is already used in Russian automotive industry. They will also encourage further research in the task-level control of the welding robotic systems, such as optimising of the weld sequence and optimal weld joint clustering in accordance with the dynamic capabilities of the robot and positioner.

Using the proposed kinematic algorithms, the designer is able to optimise special location of both a single weld and a collection of welds. In the case of a single weld, the user can directly input desired angles (8, ~) to obtain (q" q2) or to choose these coordinates in interactive mode (Fig. 2). For the collection of the welds, the positioner kinematic models are employed in Pareto-optimisation routines that yield a compromise solution, optimising the worst weld orientation. The developed routines are also utilised for welds clustering, i.e. dividing them in groups that are processed separately, for different positioner postures.

ACKNOWLEDGEMENTS The authors appreciate the financial support of the INT AS through the project 2000-757.

Recent application of ROBOMAX is the design of a robotic welding station for a new LADA car. The station includes 4 KUKA robots and a positioner with 7 d.o.f. (Fig. 3), which is actually a combination of three 2-axis positioners. The workpiece consists of two longerons, two floor connectors, an engine mounting flange and a conductor plate with fixture jigs and clamps. As follows from the result, ROBOMAX provides good capabilities for optimising weld joint orientation and coordinated control of the robot-positioner system.

REFERENCES Bolmsj6, G. (1987). A Kinematic Description of a Positioner and its Application in Arc Welding Robots. Proceedings of the Second International

Conference on Developments in Automated and Robotic Welding, paper no 2, The Welding Institute, London. Bolmsj6, G. (1999). Programming Robot Welding Systems Using Advanced Simulation Tools.

Proceedings of the International Conference on the Joining of Materials: JOM-9, Helsing0r, Denmark, 284-291. Canudas de Wit, c., Siciliano, B,

&

Bastin, G.

(1996) . Theory of Robot Control, SpringerVerlag, Berlin. Lee, H. Y., Woemle, C. & Hiller, M. (1991). A Complete Solution for the Inverse Kinematic Problem of the General 6R Robot Manipulator. ASME 1. Mechanical Design, Vol. 113,481--486. Nicoleris, G. (1990). A Programming System for Welding Robots. International Journal for the Joining of Materials, 2(2), 55-61. Pashkevich, A. P. (1996). Computer-aided design of

industrial robots and robotic cells for assembling and welding. Belorussian State University of Informatics and Belarus (Russian).

Fig. 3. Robotic station for longerons welding

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RadioElelectronics,

Minsk,