# Nipulator schematic diagram of the ma 504 koh tuck

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nipulator. Schematic diagram of the ma-
504 Koh Tuck Lye et al Table 2. Parameters for camera and LED assemblies. Camera assembly LED assembly Length (m) l~ = 0.0645 l~ = 0.0525, 15 = 0.4800 Mass (kg) mc = 0.209 mL = 0.026 (PSD) is mounted on the manipulator. The PSD consists of a LED assembly which is mounted at the tip of link 5 and a camera assembly mounted at a distance from the joint of links 4 and 1. Table 2 shows the parameters of the camera assembly and the LED assembly. Table 3 shows the parameters for each link of the parallelogram manipulator. 3.2 Free transverse vibration of Bernoulli-Euler beam In order to derive the dynamic equation of the manipulator, which comprises a flexible forearm labelled as link 5 in figure 4, it is essential to know its vibrational characteristics. Assuming link 5 is a Bernoulli-Euler beam, its vibration is governed by the Bernoulli- Euler beam equation given by 2 Et Ou2] + o = o, (1) where 0 < u < 15, P is the linear density of the beam, E is Young's modulus and I is the cross-sectional area moment of inertia of link 5. The values of p and I for link 5 are 0.244 kg/m and 6.75 x 10-11 m 4 respectively. The assumed-modes method is used in our analysis. The deflection of link 5 is given in separable form as OO v(u, t) = ~ dPi(u)qi(t), (2) i=1 where q~i(u) is the ith natural mode eigenfunctions, and qi (t) is the time-dependent gen- eralized coordinate. The above equations are solved by approximating the natural modes of flexible link 5 by the natural modes of a uniform clamped-free beam with boundary conditions given by (Craig 1981) Table 3. Parameters of each individual link. Link 1 Link 2 Link 3 Link 4 Link 5 Width (mm) 30 30 30 30 3 Height (mm) 15 15 15 15 30 Length (m) ll = 0.40 12 = 0.35 l 3 = 0.40 14 = 0.35 15 = 0.48 Centre of mass (m) Icl = 0.058 It2 = 0.090 lc3 = 0.195 1c4 = 0.173 lc5 = 0.293 Mass (kg) ml = 2.905 m2 = 1.505 m 3 = 0.877 m4 = 0.858 m5 '= 0.117 Moment of inertia 11 = 0.079 /2 = 0.031 13 = 0.023 14 = 0.019 15 = 2.24 x 10 -3 (kg/m 2 )
Control of a 2-DOF manipulator with a flexible forearm 505 Table 4. Natural frequencies and related constants of link 5. Mode i Mode 1 Mode 2 Mode 3 Mode 4 fli 15 1.87510 4.69409 7.85476 10.99554 cri 0.7341 1.0185 0.9992 1.0000 q~i(15) 2.00 -2.00 2.00 -2.00 wi (rad/s) 69.64 436.49 1540.89 3019.55 f~ (Hz) 11.08 69.47 245.24 480.58 (for mL --- 0) OV u= ° 02v 03v Plu=0 : 0, ~U : 0, 0U 2 u=15 : 0, 0U 3 u=15 = 0. (3) The natural mode shape eigenfunctions and the natural frequencies are given by (Craig 1981) where ~b i (U) = [cosh/~i u - COS ]~i u - o" i (sinh/~i u - sin/~i u)], = H /p) °5, (4) (5) 1 + cos ~i15 cosh i~i15 = O, (6) COS fill5 + cosh flil5 a/ = sin [Jil 5 + sinh [3il5 ' (7) and i = 1, . . . c~. Table 4 shows the natural frequencies and related constants for the first 4 modes of link 5. Note that the natural frequencies listed in table 4 are for link 5 without point mass at its tip. 3.3 Manipulator dynamics The method used in the dynamic modelling of the system is the Lagrange-Euler formula- tion. In this method, the computation of the kinetic and potential energy of the system is an integral part of the formulation. As the detailed derivation of the dynamic equations is very long and tedious, only the relevant equations are presented here. For more details on the derivation, please refer to appendix A.

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• Koh Tuck Lye

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