**7. Conclusion**

This work was dedicated to multi-criteria optimization problems applied to flexible robot manipulators. Initially, the model of deflection, torque and manipulability were presented for the system analyzed. Next, optimal control formalism and multi-criteria strategy were outlined. It was concluded that the effectiveness of the numerical procedure depends on the choice of the objective functions and weighting factors.

The first numerical evaluation considered torque and manipulability as performance indexes. The effect of changing the weighting coefficients was presented in Figure 3. This is a typical trend of concurrent objectives, i.e., when one performance index is improved, the other degenerates. In this context it is important to identify the contribution of each index, as shown in Figures 6 and 7.

The effect of the global optimization procedure was also discussed. This point was shown to be effective to obtain the global minimum.

The second numerical evaluation considered the end-effector error positioning and the torque as performance indexes. It was shown that there are no significant changes in the performance index while the weighting factor values are changed. It is explained by the correlation between the objectives, i.e., both indexes are directly affected by the joint torque.

Multi-Criteria Optimal Path Planning of Flexible Robots 357

Damaren, C. J., 2000, "On the dynamics and control of flexible multibody systems with

Dubowsky, S., 1994, "Dealing with vibrations in the deployment structures of space robotic

Eschenauer, H. ,Koski, J., Osyczka, A., 1990, "Multicriteria Design Optimization", Berlin,

Fu, K. S., Gonzales, R. C. and Lee, C. S. G., 1987, "Robotics: control, sensing, vision and

Khalil, W. and Gautier, M., 2000, "Modeling of mechanical systems with lumped elasticity",

Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P., 1983, "Optimization by Simulated

Latornell, D. J., Cherchas, D. B., and Wong, R., 1998, "Dynamic characteristics of constrained

Levy, A. V. and Gomez, S., 1985, "The Tunneling Method Ap-plied to Global Optimization.

Levy, A. V. and Montalvo, A., 1985, "The Tunneling Algorithm for the Global Minimization

Luenberger, D. G., 1984, "Linear and Nonlinear Programming," 2nd Edition, Addison-

Martins, J., Botto, M.A. and Sa da Costa, J., 2002, "Modeling flexible beams for robotic manipulators", Multibody System Dynamics, Vol. 7, No. 1, pp. 79–100. Matsuno, F. and Hatayama, M., 1999, "Robust cooperative control of two-link flexible

Megahed, S.M. and Hamza, K.T., 2004, "Modeling and simulation of planar flexible link

Miyabe, T., Konno, A. and Uchiyama, M., 2004, "An Approach Toward an Automated

Moulin, H. and Bayo, E., 1991, "On the accuracy of end-point trajectory tracking for flexible

Nagaraj, B.P., Nataraju, B.S. and Chandrasekhar, D.N., 2001, "Nondimensional parameters

Nagarajan, S. and Turcic, D.A., 1990, "Lagrangian formulation of the equations of motion for

Systems, Measurement, and Control, Vol. 113, pp. 320–324.

Society for Industrial and Applied Mathematics, pp. 213-244.

International Center, Sendai, Japan, pp. 5–7.

Annealing", Science, Vol. 220, pp. 671-680.

Robotics Research, Vol. 18, No. 4, pp. 414–428.

Research, Vol. 17, No. 3, pp. 211–231.

253.

Springer-Verlag.

intelligence", McGraw-Hill.

Vol. 4, pp. 3964–3969.

No. 1, 1985, pp. 15-29.

2001, Kharagpur, India.

Wesley, USA.

285–300.

closed loops", International Journal of Robotics Research, Vol. 19, No. 3, pp. 238–

systems", in: Fifth International Conference on Adaptive Structures, Sendai

Proceedings of the IEEE International Conference on Robotics and Automation,

manipulators for contact force control design", International Journal of Robotics

Numerical Optimization," (Ed. P. T. Boggs) R. H. Byrd and R. B. Schnabel, Eds.,

of Functions," The SIAM Journal on Scientific and Statistical Computing, Vol. 6,

manipulators on the basis of quasi-static equations", International Journal of

manipulators with rigid tip connections to revolute joints", Robotica, Vol. 22, pp.

Object Retrieval Operation with a Two-Arm Flexible Manipulator", The International Journal of Robotics Research, Vol. 23, No. 3, March 2004, pp. 275-291.

arms by noncausal inverse dynamic solutions", ASME Journal of Dynamic

for the dynamics of a single flexible link", in: International Conference on Theoretical, Applied, Computational and Experimental Mechanics (ICTACEM)

the elastic mechanisms with mutual dependence between rigid body and elastic

As a result, no further improvement is achieved by changing the relative importance of the objectives.

The numerical experiments conducted during the present research work lead to the following conclusions. The interdependence of the performance indexes is an important aspect that impacts the performance of the numerical method. The evaluation of different objectives that evaluates correlated information increases the computational cost but have no contribution to the optimal design. The use of several weighting factors is recommended. The nonlinear nature of the objective indexes requires a heavy exploration of the design space, aiming at finding a significant improvement for a particular combination of weighing factors. Finally, the use of a global optimization methodology is the most important contribution of this work that will distinguish the proposed methodology as a fast and accurate solver. Few iterations of the tunneling process provided an effective evolution to the global minimum.

From the author's perspective, the effective combination of different techniques is the key to obtain a high performance engineering result. Since this study deals with off-line planning, even a small improvement in the path performed by the robot may lead to expressive economic benefits. This is justified by the fact that the same movement of the robot is repeated several times during the working cycle.

The next step of this research is the evaluation of the effect of uncertainty in the design, through a stochastic approach.

As the proposed methodology is efficient to obtain an improved manipulator trajectory while dealing with the flexibility effect, joint torque and manipulability, the authors believe that the present contribution is a useful tool for robotic path planning design.
