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In recent years, establishing more and more explicit, complete and accurate dynamic models for the special category of flexible link manipulators has been a formidable challenging and still open problem in robotics research. This chapter is devoted to a methodological presentation of the application of Timoshenko beam (TB) theory (TBT) concepts to the mathematical description of flexible link robotic manipulators dynamics, as a more refined modeling approach compared to the classical Euler-Bernoulli (EB) theory (EBT) which is the conventionally adopted one. Compared with the conventional heavy and bulky rigid robots, the flexible link manipulators have their special potential advantages of larger work volume, higher operation speed, greater payload-to-manipulator weight ratio, lower energy consumption, better manoeuvrability and better transportability. However, their utilization incurs a penalty due to elastic deformation and vibration typically associated with the structural flexibility. As a consequence, the motion planning and dynamics modeling of this class of robotic manipulators are apparently made extremely complicated, as well as their tip position control. The complexity of modeling and control of lightweight flexible manipulators is widely reported in the literature. Detailed discussions can be found in (Kanoh et al., 1986; Baruh & Taikonda, 1989; Book, 1990; Yurkovich, 1992; Book, 1993; Junkins & Kim, 1993; Canudas de Wit et al., 1996; Moallem et al., 2000; Benosman et al., 2002; Robinett et al., 2002; Wang & Gao, 2003; Benosman & Vey, 2004; Dwivedy & Eberhard, 2006, Tokhi & Azad, 2008). In order to fully exploit the potential advantages offered by these lightweight robot arms, one must explicitly consider the effects of structural link flexibility and properly deal with (active and/or passive) control of vibrational behavior. In this context, it is highly desirable to have an explicit, complete and accurate dynamic model at disposal. In this chapter, we aim to present the details of our investigations concerned with deriving accurate equations of motion of a flexible link robot arm by the use of the TBT. In the first part of this work, a brief review of different beam theories and especially that of Timoshenko is given. Then, based on the TBT, the emphasis is essentially set on a detailed description of the different steps, allowing the obtaining of accurate and complete 31

Timoshenko Beam Theory based Dynamic Modeling of Lightweight Flexible Link Robotic Manipulators