Parallel Real Time Dynamics Computation and H Acceleration Feedback Control of Parallel Manipulators for Acceleration Display

Transcription

1 426 Vol. 18 No. 3, pp , 2000 H Parallel Real Time Dynamics Computation and H Acceleration Feedback Control of Parallel Manipulators for Acceleration Display Yoshihiko Nakamura 1, Katsu Yamane 1, Masafumi Okada 1 and Noriaki Komine 2 In this paper, we propose a control scheme of parallel manipulators focusing on the accuracy of acceleration on the endplate, which is an important factor when parallel manipulators are used as acceleration displays. We use two controllers dynamic controller to achieve accuracy of position and to stabilize the system, and H controller to feedback the acceleration measured on the endplate. The main problem of dynamic control is computational complexity. In order to reduce computation time for inverse dynamics, parallel processing method called multi thread programming is applied. H controller is added outside the closed loop of dynamic control to remove the vibration due to the elasticity of the links and the influence of modeling errors in the dynamic controller. Key Words: Parallel Manipulator, Dynamics Computation, Parallel Computation, H Control, Acceleration Display 1. [1] [2] Fig. 1 [3] [4] Department of Mechano Informatics, University of Tokyo 2 Shimadzu Corporation Fig. 1 Parallel mechanism with rotational/spherical joints [5] JRSJ Vol. 18 No Apr., 2000

2 H 427 [6] [7] [8] [9] [10] [12] [13] [14] [13] [15] [16] [17] H 2. 2 fi O fi A O A G W S W = O G 1 S = A 2 G fi O fi A fi G fi G = W T fi O fi G = S T fi A fi O fi G fi A 6 G W S [13] 2. 2 [13] Luh [5] Fig. 2 [13] [16] 1 A x E [8] 2 x E all [18] 3 Resolved acceleration control [5] ẍ E = ẍ Ed +K V (ẋ Ed ẋ E)+K P (x Ed x E) [13] 1 fi O Newton Euler Formulation [18] Fig. 2 Block diagram of dynamic control

3 428 x Ed K V K P 4 W S 5 fi O Newton Euler Formulation [18] fi O fi A [19] O a Fig. 3 Unifying parallel computation part b c [19] d 3 Fig. 3 Fig CPU [16] 1 PC PC JRSJ Vol. 18 No Apr., 2000

4 H 429 Table 1 Computation time for each step step time remarks (1) forward kinematics 0.92 [msec] (2) inverse dynamics 2.58 [msec] 0.43 [msec] 6 (3) torque transform 0.19 [msec] subtotal (1) 3.69 [msec] for dynamic control (4) read counter 0.60 [msec] 6 channels (5) output torque 0.30 [msec] 6 channels subtotal (2) 0.90 [msec] for I/O total 4.59 [msec] Windows95 Windows NT Solaris OS [20] OS CPU Table 1 CPU Pentium Pro 200 [MHz] OS Windows NT Windows NT WinRT [msec] 220 [Hz] [16] 3. 3 primary thread Fig. 4 Fig. 4 Scheduling of the threads CPU Event Suspend /Resume Set Event 1 D/A a b c D/A 2 a Event1 b 3 6 a b

5 430 Table 2 Comparison of computation time between parallel and serial computations with I/O without I/O serial 4.59 [msec] 3.69 [msec] parallel 1.92 [msec] 1.02 [msec] gain Event Pentium Pro 200 [MHz] CPU PC 1.92 [msec] 2 [msec] 500 [Hz] Table 2 Windows NT CPU Fig. 5 Leg of the Parallel Manipulator PID Fig. 6 [3] 70 [kg] 2 [G] Fig. 5 Table 3 Pentium Pro 200 [MHz] CPU PC Microsoft Visual C++PC D/A Windows NT WinRT Fig K P = K D = Table 3 Fig. 6 Parallel Manipulator Dynamic parameters of the leg Link Mass Inertia (kg m 2 ) [kg] I xx I yy I zz Endplate JRSJ Vol. 18 No Apr., 2000

6 H 431 Fig. 7 I/O system of the parallel manipulator Fig. 9 Endplate position in curve in xy plane Fig. 8 Endplate position in upward motion f = K P (x Ed x E)+K D d dt (x Ed x E) +K I (x Ed x E)dt 6 Fig. 10 Endplate position in upward motion when friction is considered x E x Ed J f fi = J T f 7 x E 2 [msec] K P = K D = K I = [m/s 2 ] 0.1 Fig. 8 Fig [Nm] 100 [Nm/(rad/s)] 4. 3 Fig z 5 [m/s 2 ] 5 [m/s 2 ] Fig

7 432 Fig. 11 Acceleration measured on the endplate in vertical motion Fig. 12 Bode plot of vertical direction system 5. 2 H [21] H ẍ d ẍ G m M Box Jenkins [22] Fig J H K J = WT K(I + GmK) 1 G m W SG m(i + KG m) 1 8 G = G m(1 + ) W T W S Fig. 13 Acceleration feedback system W T W S 3 K Fig [23] G D F G D = (s + 100) 2, F = 1 s ẍ d ẍ ˆẍ d ˆK Fig G Fig Fig. 11 Fig [Hz] Fig. 15 Fig. 16 JRSJ Vol. 18 No Apr., 2000

8 H [msec] Fig. 14 Measured acceleration in vertical motion with feedback B FA Fig. 15 Fig. 16 Response to sine wave input without acceleration feedback Response to sine wave input with acceleration feedback 6. [1] vol.10, no.6, pp , [ 2 ] D. Stewart: A Platform with Six Degrees of Freedom, Proceedings of the Institution of Mechanical Engineers, vol.180 1, no.15, pp , [3] 6 11 pp , [4] 6 HEXA vol.12, no.3, pp , [ 5 ] J.Y.S. Luh, M.W. Walker and R.P.C. Paul: Resolved Acceleration Control of Mechanical Manipulators, IEEE Transactions on Automatic Control, vol.25, no.3, pp , [ 6 ] M. Raghavan: The Stewart Platform of General Geometry Has 40 Configurations, Journal of Mechanical Design, Transactions of the ASME, vol.115, pp , [ 7 ] B. Mourrain: The 40 Generic Positions of a Parallel Robot, Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation, pp , [8] 6 HEXA pp.74 84, [9] 3 14 pp , [10] J.F. Kleinfinger and W. Khalil: Dynamic Modeling of Closed Loop Robots, Proceedings of the 16th International Symposium on Industrial Robots, pp , [11] Y. Nakamura and M. Ghodoussi: Dynamics Computation of Closed Link Robot Mechanisms with Nonredundant and Redundant Actuators, IEEE Transactions on Robotics and Automation, vol.5, no.3, pp , [12] M. Ait Ahmed and M. Renaud: Dynamic Modeling of Closed Chain Mehchanisms and Its Application for a 6 D.O.F. Actuated Manipulator, Proceedings of the 1st International Conference in Electronics and Automatic Control, pp , [13] vol.10, no.6, pp , [14] HEXA, vol.14, no.2, pp , [15] J. P. Merlet: Les robots paralléles. Hermés, Paris,

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