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2714 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 6, JUNE 2012bounds:|τ 1 |≤200 [N · m] and |τ 2 |≤15 [N · m]. With the endof supporting the effectiveness of the proposed controller, wehave used a squared signal whose amplitude is increased ordecreased in magnitude every 2 s. More specifically, the robottask is coded in the following desired joint positions:q d1 (t) =q d2 (t) ={ 45◦if 0 ≤ t
MEZA et al.: FUZZY SELF-TUNING PID SEMIGLOBAL REGULATOR FOR ROBOT MANIPULATORS 2715Fig. 5. Desired and actual positions 1 for the fuzzy self-tuning PID controland classical PID.Fig. 7. Applied torque to the joint 1 for the fuzzy self-tuning PID control andclassical PID.Fig. 6. Desired and actual positions 2 for the fuzzy self-tuning PID controland classical PID.origin of the state space of the closed-loop system is semiglobalasymptotically stable.With the end of getting a fair comparison, we have tuned thefixed gains of the classical PID controller in accordance withthe conditions given in Proposition 1, and we have chosen thosegains that yield the best performance in the sense of minimumrise time, without overshoot, and avoiding the possibility tosaturate the actuators. The best tuning for the fixed PID gains isobtained with= {3.5} [Nm/deg]= {0.16} [Nm/deg]= {0.96} [Nm sec/deg]= {0.35} [Nm sec/deg]= {2.09} [Nm/(deg sec)]k i2 = {0.105} [Nm/(deg sec)] .k p1k p2k v1k v2k i1Even so, we have permitted the fixed gains of the PID schemeto have an initial peak in the torque command of the elbowactuator slightly above of the torque limits during 2.5 · 10 −3s. Indeed, the control equipment is protected by software inorder to not demand, in the practice control, actions abovethe actuator torque constraints, which means that, in fact, wehave permitted that the torque 2 is saturated initially during twosample times.The experimental results are shown in Figs. 5–8. They showthe desired and actual joint positions and the applied torques forthe fuzzy self-tuning PID control and the classical PID control.From Figs. 5 and 6, one can observe that the transient responsesof the links for the classical PID control are acceptable for thefirst step, but for the following changes of the step magnitude,Fig. 8. Applied torque to the joint 2 for the fuzzy self-tuning PID control andclassical PID.Fig. 9. Proportional gain k p1 .transient responses are not really good and the accuracy ofpositioning is not satisfactory. This is due to the fact thatthe control gains are constant, and they cannot be changed inaccordance with the values of the states.The proposed fuzzy self-tuning PID controller was testedunder the same desired task. The transient responses q 1 andq 2 are shown in Figs. 5 and 6; by comparing these positionerrors with those obtained for the case of fixed gains, we see thatthe accuracy and the transient are improved by using the fuzzyself-tuning algorithm. Applied torques τ 1 and τ 2 are shown inFigs. 7 and 8; these figures show the evolution of the appliedtorques; both remain within the torque actuator limits. Theapplied torque τ 2 presents high-frequency fluctuations of smallmagnitude which are imperceptible for the mechanical system;this can be seen in the position response in Fig. 6.The improvement in the performance of the fuzzy self-tuningresponses is mainly due to the variable nature of the gainmatrices, whose evolution during the experiment can be seenin Figs. 9–14.
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