In the simulations the stability is considered by

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axis for Posture1 and Posture2, respectively. In the simulations, the stability is considered by calculating the loci of system eigenvalues when the value of K b is changed from 0.0 to -1.0. To obtain the system eigenvalues, first the model-based control system shown in Fig. 4 is re-expressed using the state variable approach, and then the eigenvalues of the system matrix are calculated. Fig. 6 shows the loci of system eigenvalues. In this figure, eigenvalues of -11±47j and -16±73j are related to the first natural frequency of the torsional vibration for Posture1 and Posture2, respectively. Fig. 6 indicates that the control system is stable on the condition of -0.9 K b 0.0. (a) Posture1 (N.F.=7Hz). (b) Posture2 (N.F.=12Hz). Fig. 5. Correlation of natural frequency with the posture. Table 1. Simulation and experimental conditions. Parameter Value Unit J m 1.362×10 - 5 J g 2.048×10 - 5 J l 2.852 Moment of inertia Posture1 Posture2 J l 1.205 kg·m 2 K s 889.6 Torsional stiffness K g 6967.3 N·m/rad C s 0.0137 Damping coefficient C g 22.553 N·m·s/rad Gear reducer Reduction ratio R g 100 Velocity loop gain K v 0.15 A/(rad/s) Integral time constant T i 1.0 s Torque constant K t 0.316 N·m/A Voltage constant K e 0.316 V/(rad/s) Phase resistance R 4.5 Phase inductance L 0.0189 H Current loop gain K c 118.84 V/A Current feedback gain K cb 1.0 Feedback gain K b 0 or -0.7 Reduced-order model Electrical part ω e 188.4 rad/s Natural frequency Damping ratio ζ e 1.0 Mechanical part Posture1 ω n 151.2 Natural Freq. Posture2 ω n 163.3 rad/s Damping ratio γ n 0.7 Posture1 R n 8.363 Inertia ratio Posture2 R n 3.256
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International Journal of Control, Automation, and Systems Vol. 1, No. 3, September 2003 267 (a) In the case of Posture1 (N.F.=7Hz). (b) In the case of Posture2 (N.F.=12Hz). Fig. 6. Loci of system eigenvalues. (a) Without model-based control. (b) With model-based control. Fig. 7. Simulation results of ω l / ω cmd in the case of Posture1. (a) Without model-based control. (b) With model-based control. Fig. 8. Simulation results of ω l / ω cmd in the case of Posture2. 3.3. Simulation of frequency response Fig. 7 and Fig. 8 show the Bode plots of the trans- fer function ω l / ω cmd . In these simulations, K b is set to -0.7 after considering simulation results of the stabil- ity. These figures indicate that the proposed model- based control equivalently increases the cut- off fre- quency of the system and the damping ratio between the reducer’s input shaft and the driven machine part. 3.4. Simulation of time response Then, the time response is calculated by the Runge-Kutta method in order to verify the suppres- sion effect on the residual vibration. Fig. 9 and Fig. 10 show simulation results. In these figures, Arm ac- cel . represents the vibration acceleration in the direc- tion of rotation at the point of L1 or L2 in Fig. 5. In these simulations, a trapezoidal velocity profile is assigned. The constant acceleration in the start phase is 1000 min -1 /28 ms. The cruise velocity is 2000 min -1 . The constant deceleration in the arrival
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