The table shows that the natural frequencies are

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The table shows that the natural frequencies are independent of the outer link orientation. V. L IMITATIONS OF THE N EW D ESIGN A desirable response for a PFBL are specified by 1) large first natural frequency . This is because we do not want the PFBL to sacrifice rigidity;
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610 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 15, NO. 4, AUGUST 1999 TABLE III R ESULTS OF THE M EASUREMENTS Fig. 7. Effect of a built in non-parallelism. 2) position invariant first natural frequency. If we allow to be the relative position between the inner and outer links (just as in the conventional manipulator), then position variation is defined as VR (5) For a good design VR ; 3) large separation between the first two natural frequen- cies. This allows the second (and higher) vibration mode(s) to be ignorable. This separation will be denoted and (6) where is the ’th lowest natural frequency. In the following, we will investigate the effect system parameters have on these measures. In addition to the previous, we need to consider the effect of nonparallelism. Although the manipulator is designed to be a PFBL, manufacturing tolerances when it is constructed may cause undesired behavior. Therefore the PFBL’s performance when it is slightly misaligned or nonparallel was investigated. The nonparallelism of the PFBL is quantified by factor defined as a fraction of coupler width. Fig. 7 defines the terms used to compute . The main effect of a nonzero is it introduces a rotation of the outer link. Fig. 7 also shows the rotation caused by vibration and a nonzero . Without much trouble, the value can be computed via (7) Equation (7) indicates that the transmitted angle caused by oscillation of the inner link is probably negligible because will typically be a very small number. A. Parameter Sweep Analysis The performance of a PFBL was investigated by deter- mining the lowest natural frequency, the frequency variation VR [see (5)] and the frequency separation [see (6)] for a TABLE IV R ANGE OF P ARAMETERS U SED IN THE A NALYSIS variety of parameters. The range of parameters was selected to be realistic values. The system parameters used in the investigation are shown in Table IV. The mass ratio range is consistent with “payload ratios” used in previous research [17]. The payload ratio is the payload weight divided by the weight of the moving structures of a robot manipulator. For an industrial rigid robot manipulator, the payload ratio varies from to [18]. Since the PFBL is supposed to be a lightweight robot, we increased the range of the ratio. In a typical robot manipulator design, the length ratio is often close to but less than one. A length ratio range from 0 to 1.5 will definitely cover all reasonable applications. The inertia ratio, , is proportional to the length ratio and the mass ratio. For example, in a slender rod, the mass moment of inertia is . In our case, we let the inertia ratio be .
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