Current Advances in Mechanical Design and Production VII. by Mohamed F. Hassan and Said M. Megahed (Eds.)

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By Mohamed F. Hassan and Said M. Megahed (Eds.)

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Additional info for Current Advances in Mechanical Design and Production VII. Proceedings of the Seventh Cairo University International MDP Conference Cairo-Egypt February 15–17, 2000

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The problems arrived from the complexity of the new developed mathematical model are solved and satisfactory results are obtained. To achieve robustness of the control algorithm, the bang-bang control is followed by PD control. Both the optimal traveling time of the bang bang control and the settling time of PD control depend on the arm structure, namely the inertia, flexibility and damping of the arm. Using finite element analysis and optimization technique, the shape of the robot arm is optimized to minimize the optimal traveling time with constraints on the settling time and end point deflection.

Then the angular displacement and angular velocity (vector y) can be found by mode summation: y=b'x, 01 b'= b0 0 5 I The resulted angular velocity is substituted in equation (6) and the new volt equation is written as" v(t) = Kb[x 2 (0) +_Tot / J + (+_qTo / c o - x3 (0)co) sin(cot) + x4 (0) cos(cot)] (8) +- 1r / K, + L(+_To - L,~ ) 8 ( 0 / K, where x2(0), x4(0) and x3(0) are the modal velocities and displacement just before the last switch and can be obtained from the solutions of equations (1) by substituting the proper time.

It is clear that, the air damping is negligible for the flexible mode, because the deflection of the flexible mode is small [18]. The state equations of the flexible arm is the same as equations (1)except matrix A which becomes: Current Advances in Mechanical Design and Production, MDP- 7 0 A= 1 0 0 i o 0 i 0 0 0 -co 2 51 (ll) where (z equals the damping coefficient (C) divided by moment of inertia of the arm (J). The solution of the Pontryagin maximum principle is a multi-switch bang-bang control but not symmetrical about the middle switch as in the previous case without damping.

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