1042296x991000 1999 ieee606 ieee transactions on

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1042–296X/99$10.00 1999 IEEE
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606 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 15, NO. 4, AUGUST 1999 a feedback signal is lost. Despite these complications, active damping can produce impressive results. Some recent research utilizes a linear control input which is a continuous, function of the dynamic state of the manipulator. These experiments have been conducted on a manipulator with a single flexible link [8], [9]–[11]. The dynamic state of the link is the sum of rigid body motion, which is directly measured, and vibrational motion, which is often estimated. Control of a single flexible link requires a high degree of accuracy in modeling manipulator dynamics and vibration estimation [11]. Introduction of multiple links compounds the difficulties in dynamic modeling and vibration estimation. Furthermore, the links are dynamically coupled such that the position of the outer links significantly affect the vibrational modes of the inner links. This coupling can be significant enough that control becomes difficult. Asada [12], Yang [13], Youcef-Toumi [14], and Park [15] proposed design rules for building rigid robot manipulators with a higher dynamic performance. The design guidelines demonstrate how to build rigid manipulators with decoupled, and/or position invariant dynamics. This paper discusses the possibility for the flexible manip- ulator’s design to affect the dynamic coupling. In turn, this possibility provides the potential for choosing a design through which the dynamic coupling is eliminated completely. If decoupling can be achieved, it may become possible to control multi-link manipulators with techniques applicable to single flexible link mechanisms. The control problem is still significant but much less difficult than the control of multi-link mechanisms. A common factor with the previous work on design rules is that they apply only to rigid manipulators. This paper discusses design rules for flexible manipulators. The paper first defines the structural dynamic problem of conventional flexible robot manipulators. Then we discuss a method for designing a flexi- ble manipulator with a simpler dynamic response. Finally, we compute and experimentally verify the variation in eigenvalues caused by manipulator position changes in the new design. II. E QUATIONS OF M OTION OF C ONVENTIONAL 2-L INK F LEXIBLE R OBOTS Consider the dynamic behavior of the two link planar manipulator with a flexible inner link and a rigid outer link that Fig. 1 shows. The inner link is numbered 1, the outer link is 2. The length of the inner and outer links are and respectively. Let both the inner and outer links have their mass lumped at the link’s end. The motor for the outer link and the payload are lumped with the link masses forming (mass on the end of the inner link) and (mass on the end of the outer link). The flexible inner link deforms like a Bernoulli- Euler beam. The vibration of the first link is assumed to have three degrees of freedom. An axial deformation is represented with . A transverse deflection is given with and a flexible
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