**1. Introduction**

338 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

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Determining the trajectory from the initial to the final end-effector positioning represents one of the most common problems in the path-planning design of serial robot manipulators. The movement is established through the specification of a set of intermediate points. In this way, the manipulator is guided along the trajectory without any concern regarding the intermediate configurations along the path. However, there are applications in which the intermediate points have to be taken into account both for path-planning and control purposes. An example of such an application is the case of robot manipulators that are used in welding operations.

In the context of industrial applications, a previous planning is justified, the so-called *off-line programming*, aiming at establishing a precise control for the movement. This planning includes the analysis of the kinematics and dynamics behavior of the system. The reduction of costs and increase of productivity are some of the most important objectives in industrial automation. Therefore, to make possible the use of robotic systems, it is important that one considers the path planning optimization for a specific task.

The improvement of industrial productivity can be achieved by reducing the weight of the robots and/or increasing their speed of operation. The first choice may lead to power consumption reduction while the second results in a faster work cycle. To successfully achieve these purposes it is very desirable to build flexible robotic manipulators. In some situations it is even necessary to consider the flexibility effects due to the joints and gear components of the manipulators for obtaining an accurate and reliable control.

Compared to conventional heavy robots, flexible link manipulators have the potential advantage of lower cost, larger work volume, higher operational speed, greater payload-tomanipulator-weight ratio, smaller actuators and lower energy consumption.

The study of the control of flexible manipulators started in the field of space robots research. Aiming at space applications, the manipulator should be as light as possible in order to reduce the launching costs (Book, 1984). Uchiyama *et al.* (1990), Alberts *et al.* (1992), Dubowsky (1994), to mention only a few, have also studied flexible manipulators for space

Multi-Criteria Optimal Path Planning of Flexible Robots 341

5 while section 6 shows numerical results. The conclusions and perspectives for future work

Different schemes for modeling of the manipulators have been studied by a number of researchers. The mathematical model of the manipulator is generally derived from energy principles and, for a simple rigid manipulator, the rigid arm stores kinetic energy due to its

A flexible link also stores deformation energy by virtue of its deflection, joint and drive flexibility. Joints have concentrated compliance that may often be modeled as a pure spring storing only strain energy. Drive components such as shafts and belts may appear distributed. They store kinetic energy due to their low inertia, and a lumped parameter

The most important modeling techniques for single flexible link manipulators can be grouped under the following categories: assumed modes method, finite element method

In the assumed modes approach, the link flexibility is usually represented by a truncated finite modal series, in terms of spatial mode eigenfunctions and time-varying mode amplitudes. Although this method has been widely used, there are several ways to choose link boundary conditions and mode eigenfunctions. Some contributions in this field were presented by Cannon and Schmitz (1984), Sakawa *et al.* (1985), Bayo (1986), Tomei and Tornambe (1988), among others. Nagaraj *et al.* (2001), Martins *et al.* (2002) and Tso *et al.* (2003) studied single-link flexible manipulators by using Lagrange's equation and the

Regarding the finite element formulation, Nagarajan and Turcic (1990) derived elemental and system equations for systems with both elastic and rigid links. Bricout *et al.* (1990) studied elastic manipulators. Moulin and Bayo (1991) also used finite element discretization to study the end-point trajectory tracking for flexible arms and showed that a non-causal solution for the actuating torque enables tracking of an arbitrary tip displacement with any

By using a lumped parameter model, Zhu *et al.* (1999) simulated the tip position tracking of a single-link flexible manipulator. Khalil and Gautier (2000) used a lumped elasticity model for flexible mechanical systems. Megahed and Hamza (2004) used a variation of the finite segment multi-body dynamics approach to model and simulate planar flexible link

Santos *et al.* (2007) proposed the computation of flexibility by means of a spring-massdamper system. According to this analogy, the first spring and damper constants are related to the joint behavior, and the following sets of spring and damper represent link flexibility.

In this work the description of the deflection related to a rigid link is proposed. It is achieved by means of an Euler-Bernoulli beam formulation and covers the case for small

The bending moment *M*, shear forces *Q* and deflections *w* for a cantilever beam subjected to

The variables and the parameters of the model are interpreted as angular quantities.

moving inertia, and stores potential energy due its position in the gravitational field.

spring model often succeeds well to consider such an effect.

manipulator with rigid tip connections to revolute joints.

deflections of a beam subject to lateral loads.

a point load *P* at the free end are given by

are given in section 7.

**2.1 Deflection** 

**2. Manipulator model** 

and lumped parameters technique.

assumed modes method.

desired accuracy.

applications. Shi *et al.* (1998) discussed some key issues in the dynamic control of lightweight robots for several applications.

As a consequence of the interest in using flexible structures in robotics, several papers regarding the design of controllers for the manipulation task of flexible manipulators are found in the literature (Latornell *et al.*, 1998), (Choi and Krishnamurthy, 1994) and (Chang and Chen, 1998).

In Tsujita *et al.* (2004), the trajectory and force controller of a flexible manipulator is proposed. From the point of view of structural dynamics, the trajectory control for a flexible manipulator is dedicated to the control of the global elastic deformation of the system, and the force control is dedicated to the control of the local deformation at the tip of the endeffector. Thus, preferably trajectory and force controls are separated in the control strategy.

Static and dynamic hybrid position/force control algorithms have been developed for flexible-macro/rigid-micro manipulator systems (Yoshikawa *et al.*, 1996). The robust cooperative control scheme of two flexible manipulators in the horizontal workspace is presented in Matsuno and Hatayama (1999). A passive controller has been developed for the payload manipulation with two planar flexible arms (Damaren 2000).

In Miyabe *et al*. (2004), the automated object capture with a two-arm flexible manipulator is addressed, which is a basic technology for a number of services in space. This object capturing strategy includes symmetric cooperative control, visual servoing, the resolution of the inverse kinematics problem, and the optimization of the configuration of a two-arm redundant flexible manipulator.

The effective use of flexible robotic manipulators in industrial environment is still a challenge for modern engineering. Usually there are several possible trajectories to perform a given task. A question that arises when programming robots is which is the best trajectory. There is no definitive solution, since the answer to this question depends on the selected performance index. Focused on industrial applications, the optimal path planning of a flexible manipulator is addressed in the present chapter. The manipulator is requested to perform a task in a vertical plane. Under this condition the gravitational effects are taken into account. Energy consumption is minimized when the movement is conducted through a suitable path. Energy is calculated by means of the evaluation of the joint torque along the path. End-effector accuracy is improved by reducing the vibration effect and increasing manipulability. The determination of the position takes into account the influence of structural flexibility. Weighting parameters are used to set the importance of each objective. The optimization scenario is composed by an optimal control formulation, solved by means of a nonlinear programming algorithm. The improvement obtained through a global optimization procedure is discussed. Numerical results demonstrate the viability of the proposed methodology.

A control formulation to determine the optimal torque profile is proposed. The optimal manipulability is also taken into account. The effect of using end-effector positioning error as performance index is discussed. As a result, the contribution of the present work is the proposition of a methodology to evaluate the influence of different performance indexes in a multi-criteria optimization environment.

The paper is organized as follows. In Section 2, model of deflection, torque and manipulability are presented. Section 3 recalls the general optimal control formulation and the performance indexes are defined. Geometrical insight about the design variables is given. Multi-criteria programming aspects such as Pareto-optimality and objective weighting are presented in section 4. The global optimization strategy is outlined in section 5 while section 6 shows numerical results. The conclusions and perspectives for future work are given in section 7.
