Consider the conditions that guarantee constant

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Consider the conditions that guarantee constant eigenvalues. Once the robot is designed, the mass, length, stiffness, and inertia ratios are given. Provided the payload remains constant, these ratios are constant. If the payload changes at discrete times, it might be possible to think of the system as a “set” of manipulators each with a different payload. Each of these manipulators however will still have a variable eigensystem because of its dependence on . This behavior is illustrated in Fig. 2 that shows the first frequency of the conventional design versus relative position for parameter values of , and , and . Fig. 1 shows the vibrational motion of the conventional flexible robot. The figure shows that when the inner link vibrates, the outer link both translates and rotates. It is this rotational motion that causes the position dependent frequencies. An easy way to understand this is to compute the “equivalent” inertia of the outer link, a concept frequently used in Lagrangian mechanics. It is this concept that leads to the new design. III. T HE N EW D ESIGN To see how to achieve the desirable behavior one can define a linkage as a general constraint mechanism. For example, suppose the manipulator is an open chain constructed of joints. Consider the motion of the body down the chain. Let the motion of body relative to body be specified via a number of coordinates such as (motor or joint position), (motor or joint velocity), and flexible velocity coordinates . If we assume are zero at the nominal rigid position and the linkage itself is massless then we can express the kinetic energy and potential energy introduced by linkage in terms of and . Using a Lagrangian principle, we can express the dynamic equations in very general terms. A complete investigation of the general equations is the subject of future work but one simple solution to them suggests designing a manipulator such that flexible vibration of link produces pure translation of mass . One way to accomplish this is to build the flexible inner link as a parallel four-bar linkage (a PFBL). It is this solution that will be investigated in the remainder of this paper. Fig. 3 shows the new design. In an ideal case, vibration of the inner link causes both sides of the PFBL to deform
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608 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 15, NO. 4, AUGUST 1999 Fig. 3. Vibrational motion of an ideal PFBL link. the same magnitude and direction causing the outer link to translate. In actuality the sides will elongate or shorten slightly causing some rotation. How much axial deformation is present determines the effectiveness of the design. The remainder of the paper discusses how close to ideal a real PFBL must behave to achieve the desired response. A. Model of the PFBL The PFBL link is constructed like a parallel four-bar linkage.
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