LINK L E N G T H C O N T R O L USING D Y N A M I C S F O R P A R A L L E L M E C H A N I S M W I T H A D J U S T A B L E LINK P A R A M E T E R S
W. Tanaka 1, T. Arai ~, K. Inoue 1, T. Takubo l, Y. Mae 2 and Y. Koseki 3 ~Department of System Innovation, Division of System Science and Applied Informatics, Graduate School of Engineering Science, Osaka University 1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japan 2Department of Human and Artificial Intelligence Systems, Faculty of Engineering, University of Fukui 3National Institute of Advanced Industrial Science and Technology
There has always been a workspace problem with parallel mechanisms. Previously we have proposed a parallel mechanism with linear passive joints to adjust link length. This parallel mechanism can achieve different workspaces by adjusting the link length. We have tried to control the link lengths of the parallel mechanism not active but passive using dynamics. This paper proposes the control algorithm of these passive linear joints, to adjust the link lengths using dynamics in the case of a 2DOF planer prototype.
Parallel mechanism, Dynamics, Adjustable mechanical parameter, Passive joint control. INTRODUCTION Parallel mechanisms have some good advantages compared with conventional articulated arms. The only drawback is its small workspace due to its in-parallel configuration. Whole of the large workspace may not always be required in one task, and the workspace may be divided into smaller sub-workspaces corresponding to individual tasks as shown in Figure 1. This is our basic idea in the paper . If each sub-workspace can be covered by a corresponding parallel mechanism that is not individuality different but a single mechanism with differently adjusted mechanical parameters, the whole workspace can be achieved by just one machine. Our idea is to cover each sub-workspace by a machine that has adjustable mechanical parameters. There are some mechanical parameters which can be adjusted, such as base plate parameters, endeffector parameters and link parameters. This paper discussed the parallel mechanism with adjustable
Figure 1 : The combinational workspace
Figure 2 : The 2-DOF planer rotary actuated parallel mechanism with adjustable link length
link parameters, since the adjustable link parameters effects the overall workspace volume more than another parameters. The link length of this parallel mechanism can be adjusted passively or actively. A mechanism with actively adjusted link length is a kind of redundantly actuated mechanism and it would not be feasible due to its high cost of many actuators. Hence, the proposed idea is to adjust the link length without actuators. There are two methods of adjusting link length; manually and automatically. We have attempted to adjust the link length automatically because manual adjustment requires the mechanism to be off. Therefore, we need to control passive joints to adjust the link length automatically. The control of serial manipulator with passive joint using the dynamics has been studied . We have applied this method to the parallel mechanism with adjustable link parameters and control the passive linear joints to adjust the link length automatically using its dynamics.
A L G O R I T H M OF LINK LENGTH CONTROL 2-DOF Planer Rotary Actuated Parallel Mechanism with Adjustable Link Length We will discuss the control algorithm using a 2-DOF planer rotary actuated parallel mechanism with adjustable link length as shown in Figure 2. This mechanism has passive linear joints on each links with a lock. When the lock is put ON or OFF, the passive linear joints can be fixed or released respectively. The joint rl 1 and r21 are active rotary joints. 01,, t~ll, 011, rtl, 021, 021' ~iJ21and r2~ show the displacement, velocity, acceleration and torque of each active rotary joint. The joint 111 and 121 are the passive linear joints and L~2, s L~2, f~2, L22, s /22 and f22 show the displacement, velocity, acceleration and force of each passive linear joint, o 0 shows the origin of the base frame and the position of the joint rl 1. L 0 shows the position of the joint r21. L~ shows the length between the joint rl 1 and r12. L2~ shows the length between the joint r21 and r22. x , • and .~ show the displacement, velocity and acceleration vector of the end-effector.
Formulization of the Algorithm of the Passive Linear Joints Control to Adjust Link Length The equations of motion of the 2-DOF planer rotary actuated parallel mechanism with adjustable link length is shown as follows.
433 Mll, M~2, M21 and M22 shows the element of the acceleration related matrix. B a and B shows the element of the matrix of the Coriolis force, centrifugal force and friction force, r a = [rll PT21]r shows a torque vector of the active rotary joint r l l and r21. fp = [f~, f2,] r shows a force vector of the passive linear joint 112 and 122. The subscript T shows the transposed matrix. Moreover, the acceleration of the active rotary joint shows ~ - [0,, 02, ]r, the acceleration of the passive linear joints shows /i/p :[L,2 L221r . When the locks on the passive linear joints are put OFF, the forces f ; = [f~ f21] r become zero. We also define that q;,d shows the desired acceleration of the passive linear joints. Eqn. is solved about the torques of the active rotary joints. qa - -M2,1M22iJ p,d - M211Bp
(- MllM211M22 + M12 ~p,d + ga _ MI1 M-1B; 21
We can input the desired acceleration of the passive linear joints to the Eqn. 2, 3, to obtain the torques and accelerations of the active rotary joints.
SIMULATION Simulations were used to investigate the algorithm of link length adjustment. First, we set up the desired trajectories of the displacements, velocities and accelerations of the passive linear joints. We estimate the torques and acceleration of the active rotary joints to obtain the desired transformation of the passive linear joints. Their torques and acceleration are given to the forward dynamics equation to estimate the acceleration of the passive linear oints. We also confirm whether the estimated acceleration of the passive linear joints is the same as the desired acceleration. If the estimated acceleration is same as the desired acceleration, we can calculate the desired link length by using their acceleration. Kinematic parameter definition
We will discuss simulations using the planer rotary actuated parallel mechanism shown in Figure 2. The parameter values are as follows: Lo=Lt~--L2~ =0.15[m] and the initial link length L~2 and L22 are 0.24[m] respectively. All of the link mass is 0.50[kg]. The center of mass for each link is the center of itself, and we assume that all of link frame axis as its principal axis of inertia are in the same direction. Gravity is ignored because this mechanism is fixed parallel to the ground. Desired Trajectory o f the Transformation o f the Passive linear joints
We set up the desired displacement of the passive linear joints L12,a and Lzz.d are 0.265[m]. Here, we also set the control time is 0.15[s], because the trajectories of the transformation of the passive linear joints are given arbitrarily, as shown in Figure 3, Figure 4 and Figure 5. Figure 3 shows the desired displacements [cm] of the passive linear joints, Figure 4 shows the desired velocities [cm/s] of the passive linear joints, Figure 5 shows the desired accelerations [cm/s 2] of the passive linear joints in each vertical axis. In these figures, the horizontal axis is time [s]. Simulation Results
We simulated the desired acceleration of the passive linear joints with Eqn. 2, 3. The estimated the torques and accelerations of the active rotary joints to realize the desired transformation of the passive
434 linear joints. Next, we give these torques and acceleration to the forward dynamics to estimate the acceleration of the passive linear joints. Figure 6 shows its results. The horizontal axis indicates time [s] and the vertical axis indicates the acceleration [cm/s 2] of the passive linear joint 112 and 122. Next, we give these torques and acceleration to forward dynamics to estimate the acceleration of the passive linear joints. Figure 6 shows its results. The horizontal axis indicates time [s] and vertical axis indicates the acceleration [cm/s z] of the passive linear joint 112 and 122. Figure 6 shows the same as Figure 5. If this mechanism is given the estimated torque and the acceleration of the active rotary joints, it is possible to calculate the desired passive linear joint displacements, velocities, and accelerations. Therefore, we can ultimately calculate the desired link length.
~a 0 Time Isl
Figure 3 The 9 desired displacement ofl12 and 122
2 Time [s]
:oo ? - - i.............
_6. i 150
Figure" 4 The desired velocity of l l2 and 122 ,....,
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100 Time Is]
'8 '0o .
~ ! .4~
. . . . . .
..... i! . . . . +--~-i
-600 ..~ Time Is]
Figure 5 The 9 desired acceleration ofll2 and 122 Figure 6 The 9 estimated acceleration ofll2 and 122 CONCLUSIONS
In this paper, we have discussed a method for link length adjustment using dynamics for the 2-DOF planer rotary actuated parallel mechanism with passive linear joints on each link. The procedure of this method is as follows: first, estimate the trajectories between the present and desired displacement, velocity and acceleration of the passive linear joints. Then, estimate the torques and accelerations of the active rotary joints in order to solve the accelerations, velocities and displacements of passive linear joints. By setting the torques and accelerations to the active rotary joints, the desired acceleration is generated on the passive linear joints. By achieving the desired trajectories of the passive linear joints, we obtain the desired link length.
References  T. Arai, et al.(2000). Parallel Mechanisms with Adjustable link length. Proc. IEEE/RSJ 2000 International Conference on Intelligent Robotics and Systems, 671-676.  H. Arai, et a1.(1991). Position Control System of a Two Degree of Freedom Manipulator with Passive Joint. IEEE Trans. Industrial Electronics, 38:1, 15-20.  H. Arai, et a1.(1991). Position Control System of a Manipulator with Passive Joints Using Dynamic Coupling. IEEE Trans. Robotics and Automation, 7:4, 528-534.