The study of coordinated manipulation of two redundant robots with elastic joints

Mechanism and Machine Theory 35 (2000) 895±909

www.elsevier.com/locate/mechmt

The study of coordinated manipulation of two redundant robots with elastic joints Zhao Jing*, Bai Shi-Xian School of Mechanical and Electronic Engineering, Beijing Polytechnic University, Beijing 100022, People's Republic of China Received 16 June 1999; accepted 30 September 1999

Abstract In this article, the problem of joint trajectory planning in the coordinated manipulation of redundant robots with elastic joint is discussed. Two optimal schemes, minimum joint torque method and minimum joint de¯ection method are proposed. Simulation analyses for the two planar 3R redundant robots with elastic joints handling a single object are also made by using these methods. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Redundant robot; Elastic joint; Coordinated manipulation; Joint trajectory planning

1. Introduction A lot of researchers have shown that most of the industrial robots have joint elasticity which is the principal source of robot elasticity [1,2]. It is because of this joint elasticity that joint elastic de¯ection will increase with the increase of robots' operation speed, which will certainly in¯uence the tracking precision of robots and thus make them dicult to ful®l high precision tasks. For a non-redundant robot, in order to keep the tracking accuracy of its end-e€ector, people usually determine the joint motions and joint control torques by minimizing the tracking error of the end-e€ector. Since this method is used to cancel out the de¯ection among every joint to achieve an ideal end-e€ector tracking accuracy and there is no limit to joint * Corresponding author. E-mail address: [email protected] (Z. Jing). 0094-114X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 9 9 ) 0 0 0 5 7 - 9

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de¯ection, the joint de¯ection may be very large, which is usually harmful to harmonic reducers and results in relatively large model errors (violates small de¯ection assumption). For a redundant robot, we can solve this problem by using its special joint self-motion to minimize joint de¯ection while satisfying its tracking accuracy [3]. The above researches mainly concentrate on a single robot. For many operation tasks in industrial production, a single robot is not competent, while the coordinated manipulation of two or more robots is usually necessary. For example, carrying a heavy or a ¯exible object, assembling complex parts and sawing, etc. In this ®eld, many scholars have made intensive researches [4±6], but they used rigid robots as study objects. When two or more robots cooperate to complete a manipulation task, the e€ects of joint elasticity on robots will become more obvious due to the kinematic and dynamic constraints between robots and grasped object. So far, less attention has been paid to the research on coordinated manipulation of redundant robots with elastic joints [7]. In order to make a robot perform more complicated tasks with high precision, more analyses must be done on the problem. Based on Refs. [3,6], this paper will make researches on joint trajectory planning of coordinated manipulation for two redundant robots with elastic joints. 2. Constraint equations When two robots grasp a common rigid object, the robots and the object constitute a closedchain system. In order to realize coordinated manipulation, certain constraint relations between the grasped object and the robots should be followed. Next, we will study these constraint relations, and develop dynamic equations of the robots and the object. Firstly, the coordinate systems of a cooperating robotic systems are set up as shown in Fig. 1, where B1 XB1 YB1 ZB1 is the base frame; C1 XC1 YC1 ZC1 and C2 XC2 YC2 ZC2 are the contact frames of end-e€ectors and object; OXO YO ZO is the object frame (O is the object's center of gravity, i.e., c.g.; coordinate axes coincide with the principal axes of the object). For convenience sake, the two robots are called master robot and slave robot, respectively, whose relevant parameters are denoted by subscript ``1'' and ``2''. Suppose that the two robots are identical, each with n joint degrees of

Fig. 1. A coordinating robotic system.

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freedom and m end-e€ector's parameters (for spatial robots, m ˆ 6; for planar robots, m ˆ 3† and the end-e€ectors can grasp the object tightly so that there is no relative motion between the end-e€ectors and the object. If the joint elasticity is taken into account, then every robot can be considered as a 2n multi-rigid-body system including n rigid links and n actuators (Dc rotors). For most cases, the joint elastic de¯ection coming from shaft, harmonic reducer and servo drive system may be modeled by a linear spring with sti€ness K [3]. Since the rotation angle of link is utilized to determine the constraint equations of a robotic system, the constraint equations of the robotic system with elastic joints are the same as that of the rigid robotic system. Due to limited space, the research results given in Ref. [6] have been quoted directly to obtain the following constraint equations.

2.1. Object and master robot XÇ O ˆ X C1 ‡ T where   P Ç XO ˆ o 2 R61 oo XÇ C ˆJ1 yÇ 1 2 R61

…1†

absolute velocity vector of the object absolute velocity vector of the end-e€ector of master robot

1

Jacobian matrix of master robot J1 2 R6n angular velocity vector of master robot's links yÇ 1 2 Rn1   @ ‰C ŠP Lyy1 2 R61 , Lˆ f @ y 1C1 O g Tˆ 0 PC1 O 2 R31

position vector of object's c.g. in the contact frame of master robot rotation transformation matrix between the contact frame of master robot and the base frame

‰C Š 2 R33

2.2. Master robot and slave robot XÇ C1 ‡ 2T ˆ XÇ C2 where XÇ C2 ˆJ2 yÇ 2 2 R61 absolute velocity vector of the end-e€ector of slave robot J2 2 R6n yÇ 2 2 Rn1

Jacobian matrix of slave robot angular velocity vector of slave robot's links

For the two coordinating robots, their acceleration equation may be expressed as

…2†

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XÈ C ˆ J yÈ ‡ JÇ yÇ

…3†

where   È C1 X 2 R121 absolute acceleration vector of the end-e€ectors of the two robots XÈ C ˆ È XC2   J 0 Jˆ 1 2 R122n Jacobian matrix of the two robots 0 J2   È Èy ˆ y 1 2 R2n1 the angular acceleration vector of the two robots' links yÈ 2 In addition, di€erentiating Eqs. (1) and (2) with respect to time and rearranging them, we obtain the acceleration relationship between the end-e€ectors and the object   È O ÿ TÇ X …4† XÈ C ˆ È X O ‡ TÇ where,

  PÈ O 2 R61 XÈ O ˆ oÇ O

is the absolute acceleration vector of the object. The above derived results show that Eqs. (1), (2) and (4) are the constraint equations of the robot±object system. When these constraint equations are satis®ed, the two robots can perform coordinated manipulation. Generally speaking, in order to perform coordinated manipulations, the joint motion of the master robot should be determined ®rstly by using the constraint equations between the object and the master robot according to the desired motion of the object. Then, the corresponding joint motion of the slave robot can be decided by using the constraint equations between the master robot and the slave robot. If both the robots are redundant, i.e., n > 6, Eq. (4) shows that the joint velocity of the master robot yÇ 1 has an in®nite number of solutions for the speci®ed object's motion XÇ O and o : The joint motion of the master robot can be solved uniquely by ®rst redundancy resolution based on a performance criterion (minimum joint velocity or singularity avoidance). Then, taking the result into Eq. (2) to make second redundancy resolution, we can ®nally determine the coordinated motion of the slave robot. Obviously, it is a kinematically coordinated motion. It is seen from above, like a single redundant robot, that the two coordinating redundant robots not only can accomplish operating tasks, but also can get optimal joint con®gurations by a criterion to improve their performance. 3. Dynamics equations Like the coordinated manipulation of rigid robots, the dynamic coordination should also be considered for the coordinating robots with elastic joints besides the above kinematic coordination. The dynamic coordination relationship between the robots and the object is decided by their dynamic equations. The dynamic equation of the coordinating robots with

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elastic joints can be easily derived by the dynamic equation of a single robot with elastic joints [3] and rigid coordinating robots [6]   DyÈ ‡ H ‡ G ‡ J T F ‡ nK nyy ÿ N ÿ1 q ˆ 0   IR qÈ ÿ N ÿ1 K nyy ÿ N ÿ1 q ‡ m qÇ ˆ t

…5†

where q, qÇ , qÈ 2 R2n1 D 2 R2n2n H 2 R2n1 G 2 R2n1 IR ˆ diag‰Ii Š 2 R2n2n K ˆ diag‰Ki Š 2 R2n2n n ˆ diag‰ni Š 2 R2n2n N ˆ diag‰Ni Š 2 R2n2n m ˆ diag‰mi Š 2 R2n2n t 2 R2n1   F1 2 R2m1 Fˆ F2

rotation angle, angular velocity and acceleration vectors of rotors symmetric inertia tensor of the two rigid robots (including stators) coriolis and centrifugal force vector gravity vector of the two robots inertia tensor of the rotors sti€ness matrix of the joints transmission ratio matrix of the gear reducers transmission ratio matrix of the harmonic reducers viscous damping coecient matrix driving torque vector of the joints force vector applied to the object by the end-e€ectors of the two robots

When an object is held rigidly by the two robots, its dynamic equation can be written as      G M 0 PÈ O ‡ ˆ F1 ‡ F2 oÇ O  Io oO 0 I oÇ O

…6†

where M ˆdiag‰mO , mO , mO Š 2 R33 I ˆdiag‰Ix , Iy , Iz Š 2 R33 PÈ O 2 R31 Ç O 2 R31 o Gˆ‰0, mO g, 0Š 2 R31

mass matrix of the object principal moments of inertia of the object acceleration vector of the object's c.g. angular acceleration vector of the object about its principal axes gravity vector of the object

Since the motion of the object is always known during coordinated manipulation, the left-side of Eq. (6) can be calculated. Now, we denote the left-side of Eq. (6) as R, and rewrite Eq. (6) as R ˆ WF

…7†

where R 2 R61 are the resultant forces applied to the object and W 2 R612 is the load matrix. The load matrix W can be determined uniquely once the object's orientation and relative position between the object's c.g. and contact point are known. Combining Eqs. (5) and (7) results in the dynamic coordination equations of the two robots with elastic joints. In Eq. (7), since the number of the unknown is always greater than that of the equations, the forces

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applied to end-e€ectors have in®nite solutions for the coordinated manipulations of both redundant and non-redundant robots. We call this situation as dynamic redundancy. Similar to kinematic redundancy, the dynamic redundancy can also be resolved using a performance criterion. Compared with the dynamic equation of rigid coordinating robots, it is obvious that the dimensionality of the dynamic equation of coordinating robots with elastic joints has increased from 2n to 4n, thus the problems of the joint trajectory planning and control will be more complicated.

4. Joint trajectory planning To determine the joint torques of the robots, we can solve the ®rst part of Eq. (5) for q, and then di€erentiate q twice with respect to time to get qÇ and qÈ : Finally, substitute q, qÇ and qÈ into the second part of Eq. (5) and rearrange it, which leads to [3] ÿ …†
ÿ1 È ‡ FÈ J ‡ IR NnyÈ Ç


È yÈ ‡ H È ‡G IR N …nK † Dyy 4 ‡ 2Dy y ‡D ÿ
ÿ



ÿ1 ÿ1 Ç ‡ FÇ J ‡ m NnyÇ ˆ t Ç yÈ ‡ H Ç ‡G ‡…Nn † DyÈ ‡ H ‡ G ‡ FJ ‡ m N …nK † Dyy ‡ D T Ç FÈ J ˆ JÈ T F‡2JÇ T FÇ ‡ J T F: È where FJ ˆ JT F, FÇ J ˆ JÇ F‡JT F, Di€erentiating Eq. (3) twice with respect to time gives


… † … † Ç


Jyy 4 ‡ 3Jy y ‡ 3JÈ yÈ ‡ J yÇ ÿ X C4 ˆ 0

…8†

…9†

Thus, Eqs. (7)±(9) constitute the coordination equations of the two coordinating redundant robots with elastic joints. By the minimum load method, minimum torque method or multiple criterion method proposed in Ref. [6], the load distribution and joint trajectory can be optimized. Using the minimum load method which distributes the load in least norm sense, Eq. (7) gives F ˆ W ‡R

…10†

Since W ‡ and R depend only on the object's motion, they can be regarded as constants. On the basis of this load distribution, the joint trajectory planning will be optimized in the following sections by using joint torque and joint elastic de¯ection as performance criteria. 4.1. Minimum joint torque method (called as MT method) To improve the calculation stability of the algorithm, use multiple criteria to make an objective function as follows  T   … † … † Z ˆ C1 t T t ‡ C2 y 4 y4 …11† In consideration of the kinematic constraints, the problem of trajectory planning with minimum joint torques is expressed as

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901

 T   … † … † Min Z ˆ C1 t T t ‡ C2 y 4 y4

…12†




… † Ç


Subject to Jyy 4 ‡ 3Jy y ‡ 3JÈ yÈ ‡ J yÇ ÿ X…4 † ˆ 0

…13†

Introducing Lagrange multipliers l ˆ‰l1 , l2 , . . . ,lm ŠT 2 Rm1 , the new objective function is  T    


… † … † … † Ç


Z 0 ˆ C1 t T t ‡ C2 y 4 …14† y 4 ‡ l T Jyy 4 ‡ 3Jy y ‡ 3JÈ yÈ ‡ J yÇ ÿ X …4 † The above optimization problem is equivalent to  T    


… † … † … † Ç


y 4 ‡ l T Jyy 4 ‡ 3Jy Min Z 0 ˆ C1 t T t ‡ C2 y 4 y ‡ 3JÈ yÈ ‡ J yÇ ÿ X …4 †

…15†

In Eq. (15), except y …4† , the remaining parameters are constant for the speci®ed X …4† , y , yÇ , yÈ


and y : To minimize Z', the following conditions must be satis®ed: 8 > @Z 0 > > < È ˆ0 @y …16† > @Z 0 > > ˆ0 : @ll Solving Eq. (16), we can determine the joint motion with minimum joint torques     … † …4 † È yÈ ÿ J


yÇ ‡ I ÿ J ‡ J BA Ç


ÿ 3 J ÿ 3 Jy y 4 ˆ J‡ X y B B

…17†

where T T ÿ1 2 R2n2m J‡ B ˆBJ ‰JBJ Š 2n2n I2R Bˆ Aˆ

C1, C2

weighted generalized inverse of Jacobian identity matrix f2C1 ‰I R N…nK†ÿ1 DŠ2 ‡2  4:5  10ÿ7 C2 Igÿ1 2 R2n2n È ‡ FÈ J †‡I R NnyÈ ‡ Ç


È yÈ ‡ H È ‡G ÿ2C1 ‰IR N…nK†ÿ1 DŠT ‰IR N…nK †ÿ1 …2Dy y ‡D


Ç È ÿ1 ÿ1 È Ç Ç mN…nK† …Dyy ‡ Dy ‡ H ‡ G‡ …Nn† …Dy ‡ H ‡G‡FJ † ‡m 2n1 Ç ÇFJ † ‡m mNny Š 2 R weight factors

It is worth pointing that the constant 4.5  10ÿ7 in B is an adjusting factor which can correct the magnitude order di€erence of the di€erent sub-objective functions. It is equal to the element of diagonal matrix ‰I R N…nK†ÿ1 Š2 : 4.2. Minimum joint elastic de¯ection method (called as MD method) In the ®rst of Eq. (5), nyy ÿ N ÿ1 q is the joint elastic de¯ection of the robots and we denote it with P: Thus, P can be expressed as   ÿ1 …18† P ˆ ‰nK Š DyÈ ‡ H ‡ G ‡ JT F T If taking C1 PT P ‡ C2 yÈ yÈ as an objective function, JyÈ ‡ JÇ yÇ ÿ XÈ C ˆ 0 as a constraint equation,

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through the same derivation as the above, the joint motion with minimum joint elastic de¯ections can be achieved     ‡ Ç È …19† yÈ ˆ J ‡ B X ÿ J y ‡ I ÿ J B J BA where T T ÿ1 2 R2n2m weighted generalized inverse of Jacobian J‡ B ˆBJ ‰JBJ Š 2n2n identity matrix I2R ÿ2 2 ÿ1 2n2n Bˆ‰2C1 …nK† D ‡2C2 IŠ 2 R Aˆÿ2C1 …nK†ÿ2 D‰H ‡G‡J T FŠ 2 R2n1 C1, C2 weight factors

5. Simulation examples In this section, we will make simulation researches on the coordinated manipulation of two planar 3R robots with elastic joints holding a common object shown in Fig. 2. Suppose that the hard point contacts between the object and the end-e€ectors. Thus, only forces are transmitted from the end-e€ectors to the object. The gravity of the system is also neglected. Since only position parameters are considered here, i.e., n ÿ m ˆ 1, the two planar 3R robots can be regarded as redundant robots. The rotation angles of the two redundant robot links are denoted as: y 1 ˆ ‰y11 , y12 , y13 ŠT and y 2 ˆ ‰y21 , y22 , y23 ŠT , respectively. In terms of Eq. (4), the robots' motion is expressed as

Fig. 2. Two planar 3R robots holding a common object.



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PÈ ÿ TÇ XÈ C ˆ È O Ç PO ‡ T



903

…20†

where    ÿa sin j a cos j j ‡ j2 , j is rotation angle of the object …see Fig: 2† TÇ ˆ ÿa cos j ÿa sin j 

If the object's motion is known, the exerted resultant force R can be represented as iT h R ˆ mO P Ox , mO P Oy ‡ mO g, Iz j

…21†

Load matrix is 2

3 1 0 1 0 5 W ˆ 40 1 0 1 ÿa cos j a sin j a cos j ÿa sin j

…22†

Further, obtain the M±P generalized inverse of W 2

1 62 6 6 60 6 6 ‡ W ˆ6 61 6 62 6 4 0

0 1 2 0 1 2

cos j ÿ 2a sin j 2a cos j 2a sin j ÿ 2a

3 7 7 7 7 7 7 7 7 7 7 7 5

…23†

In simulation calculations, mO ˆ 1 (kg), 2a ˆ 0:5 (m), the object motion is speci®ed by a ninthorder polynomial 8   > 70 9 315 8 540 7 420 6 126 5 > > > < POx ˆ l0 ‡ 0:1 T 9 t ÿ T 8 t ‡ T 7 t ÿ T 6 t ‡ T 5 t   > 70 9 315 8 540 7 420 6 126 5 > > > : POy ˆ l0 ‡ 0:3 T 9 t ÿ T 8 t ‡ T 7 t ÿ T 6 t ‡ T 5 t where l0 can be decided by the initial position of the object; T is the simulation time.

…24†

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Table 1 Structural parameters and the initial con®gurations Link length (m)

Master robot Slave robot

Link mass (kg)

Initial con®guration (deg)

l1

l2

l3

m1

m2

m3

y1

y2

y3

0.5 0.5

0.5 0.5

0.5 0.5

1 1

1 1

1 1

ÿ45 60

45 ÿ60

60 ÿ120

Fig. 3. Joint con®guration. (a) The MT method and (b) the MS method.

Fig. 4. The total joint torques of the two robots.

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Fig. 5. The total joint de¯ections of the two robots.

T ˆ 0:4 (s), C1 ˆ 1, C2 ˆ 0:1, Ki ˆ 100 (N-m/rad), Ii ˆ 0:002 (kg m2), mi ˆ 0:01 (N-m s/rad), Ni ˆ 40, ni ˆ 1:2, the structural parameters and the initial con®gurations of the two robots are shown in Table 1. First, the MT method and the MS method …C1 ˆ 0† (Minimum Snap method) are utilized to

Fig. 6. The total joint torques of the two robots.

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Fig. 7. The total joint de¯ections of the two robots.

make simulation researches, whose results are shown in Figs. 3±7. The total torques of the two robots are given in Fig. 4. Compared with MS method, the maximum of joint torques for the MT method has decreased by about 10%. Further, comparison of every joint torque (omit ®gures) indicates that the torque of joint 1 is a main factor to in¯uence the total joint torques and the MT method can decrease the torque of joint 1 considerably. Meanwhile, the result in Fig. 5 shows that the total joint de¯ection of the MT method is smaller than that of the MS method. In addition, according to de¯ection curves of every joint (omit ®gure), we can see that the di€erence of joint 1 between the two methods is largest. The above analyses seem to indicate that for the two indexes Ð joint torque and joint de¯ection Ð the MT method is better than the MS method. However, it is not always true. A lot of simulation calculations show that with the decrease in the acceleration of the robotic system, the MS method will be better than the MT method, which can be seen from the simulation results Figs. 6 and 7. In this case, T ˆ 1 (s), other condition is the same as before. Why cannot the MT method give the minimum joint torques as expected? The principal reason is that the MT method is a local optimization method. The joint motions determined by Eq. (17) cannot give the minimum joint


torques globally, but only give the minimum joint torques locally for the decided y , yÈ , yÇ and y :


È Ç Since the values of y , y , y and y for the MT method and the MS method are di€erent over the entire motion except their initial values, in view of the global characteristic of joint torque, it is possible that the MT method is better than the MS method. In Eq. (8), the joint torques consist of two parts, the ®rst has y …4† terms and the second has the remaining terms. The MT method can reduce the ®rst part to a minimum by canceling out each other among y …4† terms. But due to the increase of y …4† , the second part will increase to some extent, while the MS method can reduce the two parts to some extent by minimizing y …4† : When the system

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Fig. 8. Joint con®guration. (a) The MD method and (b) the MA method.

acceleration is greater, the MT method is better since the ®rst part is a principal factor to a€ect the joint torques. When the system acceleration is smaller, the MS method is better since the second part is a principal factor to a€ect the joint torques. Next, Figs. 8±10 present the simulation results for the MD and the MA method …C1 ˆ 0† (Minimum Acceleration method). It can be seen from them that the MD method reduces the joint de¯ections but increases the joint torques. And the di€erences of joint torque and joint de¯ection for two methods mainly come from the joint 1, which corresponds to the MT method and the MS method. It means that the essence of the dynamic index methods (MT and

Fig. 9. The total joint torques of the two robots.

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Fig. 10. The total joint de¯ections of the two robots.

MD methods) is to optimize the dynamic index of joint 1 as much as possible. In addition, comparing the MT method with the MD method, we ®nd that the former can obtain smaller joint torques and the latter can obtain smaller joint de¯ections. It is worth pointing, in above researches, that the performance indexes of the master and slave robots are completely same. If we have di€erent kinematic or dynamic requirements for the two robots, for example, requiring the master robot to reduce its joint torque or joint de¯ection as much as possible, a weight matrix can be used.

6. Conclusions Putting together, we conclude: for the coordinated manipulation of the two redundant robots with elastic joints, the dynamic index methods, the MT method, and the MD method can be utilized to get smaller joint torques and de¯ections, respectively, when the system acceleration is greater; the kinematic index methods, the MS method, and the MA method which have a much better computational stability should be utilized when the system acceleration is greater.

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[3] Zhao. Jing, Bai. Shi-Xian, The e€ects of redundancy resolution on joint elastic de¯ection of redundant robot, ROBOT 20 (1) (1998) 37±42. [4] J.Y.S. Luh, Y.F. Zheng, Constrained relations between two coordinated industrial robots for motion control, Int. J. Robotics Research 6 (3) (1987) 60±70. [5] F.T. Cheng, Control and simulation for a closed-chain dual redundant manipulator system, Journal of Robotics Systems 12 (2) (1991) 119±133. [6] Zhao Jing, Bai Shi-Xian, Load distribution and joint trajectory planning of coordinated manipulation for two redundant robots, Mechanism and Machine Theory 34 (8) (1999) 1155±1170, in press. [7] K.P. Jankowski, Inverse dynamics control of multiple robot arms with ¯exible joints, IEEE Int. Conf. on Robotics and Automation, 1993, pp. 996±1003.