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This chapter focuses on the optimal shape design of flexible manipulators without considering control or sensing effects; that is, traditional design problems instead of mechatronic design problems. Various design models, problem formulations, and solution procedures will be investigated here, from Euler models to Timoshenko theory, from classic beam design to multiple constrained arm shape optimization, from an analytical approach with iterative steps to numerical methods with search procedures. This chapter presents not only the state of the art in optimization of flexible manipulators, but also the foundation for the mechatronic design to be discussed in the next chapter.

In this chapter a comprehensive study is presented on the influence of rotatory inertia, shear deformation, and tip load on the vibration behavior of one-link flexible manipulators. Three types of dynamic models, the Euler–Bernoulli model, the Euler–Bernoulli model with rotary inertia, and the Timoshenko model, have been established and used for the analysis of modal frequencies and vibration modes. A complete analysis of natural frequencies and modal shape functions of flexible manipulators has been undertaken. Explicit solutions of the asymptotic behavior of high-order modal frequencies and vibration modes are given and verified with numerical results. Both asymptotic analysis and numerical results indicate that the effect of shear deformation cannot be ignored for high-order vibrations or for shorter manipulators. It has also been found that, in certain cases, the tip load has a profound effect on the vibration behavior of flexible manipulators. Finally, the experiments conducted for modal frequency identification show excellent agreement between the analytical prediction and the experimental results.

The construction of lightweight manipulators with a larger speed range is one of the major goals in the design of well-behaved industrial robotic arms. Their use will lead to higher productivity and less energy consumption than is common with heavier, rigid arms. However, due to the flexibility involved with link deformation and the complexity of distributed parameter systems, modeling and control of flexible manipulators still remain a major challenge in robotic research. A compromise between modeling costs and control efficiency for real-time implementation is inevitable. The interdependency of subsystems results in a local optimal performance in the traditional design scheme. An important research topic in flexible manipulator design is the pursuit of better system performance while avoiding model-intensive or control-intensive work. This problem can be solved using the proposed mechatronic design approach. It treats the mechanical, electrical, and control components of a flexible manipulator concurrently. The result is a global optimal design with an explicit link shape and controller parameters that result in the control problem and modeling accuracy no longer being critical for obtaining desired performance, with the coupling effects among subsystems being taken into account automatically. Dynamics of flexible manipulators with shear force and rotary inertia are derived, and state-space equations with the integration of DC motor dynamics are developed as a theoretical basis for mechatronic designs. Case studies based on LQR formulas and H ∞ are considered. The beam shape and controller parameters are obtained using an adaptive iterative algorithm with the accommodation of various geometrical constraints. Also, different output feedback strategies are investigated to evaluate the impacts of various controller structures.

In this chapter, both the Euler–Bernoulli and Timoshenko beam theories are used to construct the dynamic models of single-link flexible manipulators. We start from the nonlinear models and then, for the purpose of vibration analysis, mechatronic design, and control synthesis of flexible manipulator systems, linearized models are derived by neglecting all deformation terms with orders higher than first order. Since the resulting dynamic models, described in terms of partial differential equations, are of infinite dimension and cannot be used directly for controller designs based on the finite dimensional model, various approximate finite dimensional modeling methods for flexible manipulators are investigated in this chapter, including the assumed modal method, finite difference method, and finite element method.

This chapter considers the early research on flexible manipulators, and then goes on to summarize more recent work on the subject.