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2716 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 6, JUNE 2012VII. CONCLUSIONFig. 10. Proportional gain k p2 .Fig. 11. Integral gain k i1 .In this paper, we have proposed a fuzzy adaptation schemefor tuning the proportional, integral, and derivative statedependentgains of a PID controller for robot manipulators.Moreover, a semiglobal asymptotic stability proof for the proposedfuzzy self-tuning PID controller for robot manipulators ispresented. The proposed approach allows to consider importantpractical features in real robots, such as achievement of desiredaccuracy and avoidance of working of the actuators’ torques beyondtheir capabilities. The performance of the proposed fuzzyscheme has been verified by means of real-time experimentaltests on a two-degree-of-freedom direct-drive robot arm.In summary, we have extended the stability analysis presentedin [34] from PID controllers with fixed gains to PIDcontrollers with variable gains for robot manipulators. Furthermore,we have applied such an analysis in order to self-tune,via fuzzy logic, the proportional, derivative, and integral gainmatrices. The new resulting control scheme is superior to thatshown in [34] due to the use of variable gain matrices and isalso superior to control schemes introduced in [18] and [25],because the new proposed controller does not employ explicitlythe knowledge of the robot dynamics model. Our result is asynergy of [34] and [25].Fig. 12. Integral gain k i2 .Fig. 13. Derivative gain k v1 .Fig. 14. Derivative gain k v2 .REFERENCES[1] V. Mummadi, “Design of robust digital PID controller for H-bridge softswitchingboost converter,” IEEE Trans. Ind. Electron., vol. 58, no. 7,pp. 2883–2897, Jul. 2011.[2] R. J. Wai, J. D. Lee, and K. L. Chuang, “Real-time PID controlstrategy for maglev transportation system via particle swarm optimization,”IEEE Trans. Ind. Electron., vol. 58, no. 2, pp. 629–646,Feb. 2011.[3] K. Kim, P. Rao, and J. A. Burnworth, “Self-tuning of PID controller for adigital excitation control system,” IEEE Trans. Ind. Appl., vol. 46, no. 4,pp. 1518–1524, Jul./Aug. 2010.[4] R. Muszynski and J. Deskur, “Damping of torsional vibrations in highdynamicindustrial drives,” IEEE Trans. Ind. Electron., vol. 57, no. 2,pp. 544–552, Feb. 2010.[5] M. A. S. K. Khan and M. A. Rahman, “Implementation of a waveletbasedMRPID controller for benchmark thermal system,” IEEE Trans.Ind. Electron., vol. 57, no. 12, pp. 4160–4169, Dec. 2010.[6] R. E. Brown, G. N. Maliotis, and J. A. Gibby, “PID self-tuning controllerfor aluminum rolling mill,” IEEE Trans. Ind. Electron., vol. 29, no. 3,pp. 578–583, May/Jun. 1993.[7] S. Skoczowski, S. Domek, K. Pietrusewicz, and B. Broel-Plater, “Amethod for improving the robustness of PID control,” IEEE Trans. Ind.Electron., vol. 52, no. 6, pp. 1669–1676, Dec. 2005.[8] L. Guo, J. Y. Hung, and R. M. Nelms, “Evaluation of DSP-based PIDand fuzzy controllers for DC–DC converters,” IEEE Trans. Ind. Electron.,vol. 56, no. 6, pp. 2237–2248, Jun. 2009.[9] J. Moreno-Valenzuela and R. Campa, “Two classes of velocity regulatorsfor input—Saturated motor drives,” IEEE Trans. Ind. Electron., vol. 56,no. 6, pp. 2190–2202, Jun. 2009.[10] S. Arimoto and F. Miyazaki, “Stability and robustness of PID feedbackcontrol for robot manipulators of sensory capability,” in Robotics. ResearchFirst Int. Symp., M. Brady and R. P. Paul, Eds. Cambridge, MA:MIT Press, 1984, pp. 228–235.[11] CIRSSE Document 54 J. T. Wen and S. Murphy, PID Control for RobotManipulators1990.[12] P. Roco, “Stability of PID control for industrial robot arms,” IEEE Trans.Robot. Automation, vol. 12, no. 4, pp. 606–614, Aug. 1996.[13] Z. Qu and J. Dorsey, “Robust PID control robots,” Int. J. Robot. Autom.,vol. 6, pp. 228–235, 1991.[14] R. Ortega, A. Loria, and R. Kelly, “A semiglobally stable output feedbackPI 2 D regulator for robot manipulators,” IEEE Trans. Autom. Control,vol. 40, no. 8, pp. 1432–1436, Aug. 1995.