Weiling Zhang， Qinjun Du， Zhengyang Zhao， Chuanming Song，Wei Ding and Yonggang Luo

（School of Electrical and Electronic Engineering, Shandong University of Technology,Zibo 255049, Shandong, China）

Abstract: To address the problem of resonance in the control of a robot arm, a resonance suppres?sion strategy is proposed for a single?joint humanoid robot arm based on the proportional?resonant(PR) controller. First, an arm joint model of the humanoid robot is established. Then the influence of resonance frequency on the performance of the control system with the robot arm is analyzed. The voltage fluctuation of the drive motor caused by the changes in arm motion is recog?nized as the disturbance of the current loop. The PR controller has the characteristic of disturb?ance rejection at a specific frequency. The output fluctuation of the driving system caused by the change of arm motion state at the resonance frequency is suppressed. Therefore the output current of the inverter will not be affected by the vibration of the arm at the resonance frequency. Finally,the control system is verified by MATLAB/Simulink simulation. The simulation results demon?strate that the control strategy for the humanoid robot arm based on the PR controller can sup?press the resonance of the arm effectively, improving the dynamic performance and system stability.

Key words: humanoid robot arm；resonance；proportional?resonant(PR) controller；disturbance rejection

Humanoid robots have human appearance characteristics, as well as the functions corres?ponding with their appearance features. Hu?manoid robots are widely applied to dangerous operations, medical treatment, service business,and so on. It can replace human to handle many operations in various tasks without changing the work environment, so it has a broad application prospect. The performance of humanoid robot arm will affect the completion of a task. Harmon?ic reducer is often used in humanoid robot arm to connect motor with arm link. The inherent flexibility in harmonic driver produces resonance frequencies. When control frequency equals to the resonance frequency of arm, the resonance phe?nomenon will occur. The resonance not only harms the structure of the joint and affects its service life but also leads to controlled quantities to oscillate in the robot arm control system. The operation performance of the humanoid robot arm and the stability of the system are seriously affected. Therefore, the resonance suppression strategy is indispensable in motion control of hu?manoid robots.

A number of studies on the vibration of ro?botic arms have been performed at present[1?3]. In the motion control field, many resonance sup?pression control methods such as notch filter, res?onance ratio control and self resonance cancella?tion and so on are proposed. Yang et al.[4]used a notch filter to attenuate the amplitude of spe?cified frequency without affecting other frequen?cies. The resonance of industrial robots is effect?ively suppressed. Ref. [5] used the zero phase er?ror notch filters in order to suppress the vibra?tion phenomenon which is generated by the res?onant frequency and the nonlinear interference force from the other joints of robot. It realizes the fast accurate robot motion control without vibration phenomenon and overshoot. However,improper design parameters of notch filter will lead to phase angle advance or lag of other fre?quency signal near the resonant frequency. Be?sides, the resonance of two different frequencies will be inducted into the system while the mech?anical resonance is suppressed by the notch filter.Hence the mechanical resonance cannot be elim?inated completely. An adaptive notch filter has the advantage over a notch filter because an ad?aptive one can automatically adjust the notch frequency to the resonant frequency[6?7]. However,the parameters of adaptive notch filter are diffi?cult to meet the requirements when high fre?quency resonance occurs. The deviation between the resonance frequency and the resonant peak frequency will lead to the failure of the proposed method. Ref. [8] adopted slow resonance ratio control for vibration suppression and disturb?ance rejection in the torsional system. The phase margin of the system decreases when the nomin?al inertia of the motor is greater than the actual inertia of the motor. And large deviation will cause system instability. It was possible to design a feedback controller which not only broadens the bandwidth but also improves the robustness against the resonance mode parameters using the self resonance cancellation(SRC) method[9]. The primary resonance mode is cancelled by using SRC and the second resonance can be reduced by adjusting the mechanical parameter. However,because SRC does not take modeling errors into consideration, it is difficult to be applied to a hu?manoid robot which changes moment of inertia,centrifugal force, coriolis force, gravity, frictional force according to the postual changes. Moreover,SRC has a problem that a steady?state error arises due to load side disturbance torque. A ro?bust control method for two inertia systems was proposed based on SRC and the self resonance cancellation disturbance observer (SRCDOB)[10].The proposed robust resonance suppression con?trol method is applied to a humanoid robot con?trol system and experimental results demon?strate the effectiveness of the proposed robust.Nevertheless, the control algorithm has many set?ting parameters and the design procedure is com?plicated. A lot of debugging and trial and error is required in the actual simulation process. From above analysis, the previous methods fail to solve the resonance of humanoid robot arm caused by joints flexibility.

In order to suppress the resonance of the ro?bot arm and achieve a good control effect, this paper proposes a resonance control strategy for humanoid robot arm based on proportional?res?onant(PR) controller[11?13]. The gain of the PR controller at the resonance frequency is infinite,which can guarantee the zero?tracking?error for sinusoidal reference. Moreover, there is no atten?uation outside that frequency. As a result, it can be used as a notch filter so that it suppresses the vibration at a specific frequency. The PR control?ler has the characteristic of disturbance rejection at the specific frequency without affecting the performance at other frequencies. Therefore, the control frequency is not equal to the arm reson?ant frequency so that the self?excited vibration will not be excited. The proposed strategy im?proves performance and ensures the stability of the system. The scheme can suppress the reson?ance of the humanoid robot arm and has a good control effect.

1 Analysis of Resonance for Humanoid Robot Arm

1.1 Arm joint modeling

As to humanoid robots, a high load?to?weight ratio is a key requirement. The compact high?gear reduction, such as the harmonic redu?cer is frequently used in the driving system for the sake of high load?to?weight performance. The flexibility of the robot arm joint increases due to the harmonic driver. Also, joint flexibility plays a role of protection to the robot arm and extends its service life.

A simplified model for the single?joint hu?manoid robot arm is established based on the model of a flexible joint robot proposed by Spong[14], as shown in Fig. 1. The joint flexibility is modeled as a linear spring with stiffness, where K is the joint stiffness and N is the gear ratio.

Fig. 1 Simplified model for humanoid robot arm

The mathematical model of humanoid robot arm joint is established based on the momentum conservation of the system and the Lagrange equation[15]. Its dynamical equation is

Here, a brushless DC motor is used as the driving motor for the humanoid robot arm joint,and the voltage equations can be described as

1.2 Resonance mechanism analysis

The block diagram of the humanoid robot arm joint is shown in Fig. 2.

Fig. 2 Block diagram of humanoid robot arm joint

The closed?loop transfer functions from the motor torque to the motor and arm angle, re?spectively, for the control structure, demonstra?ted in Fig. 2, are given as

The transfer function is shown in Eq. (8)contains a pair of imaginary conjugate poles and a pair of imaginary conjugate zeros. The imagin?ary conjugate poles in Eq. (9) are the same as the ones in Eq. (8). The introduced imaginary conjugate poles and zeros will reduce the per?formance of humanoid robot arm motion system.It could have a strong influence on the control quality of the drive system and the practical op?eration effect. Here, the conjugate zeros intro?duce the anti?resonance frequency(ARF), and the conjugate poles introduce the natural torsion?al frequency(NTF) of the system, namely, the resonance frequency. The resonance frequency of the motor portion and the anti?resonance fre?quency of the arm portion are defined as follows

Fig. 3 Block diagram of the closed?loop system

Bode plot of the open?loop and closed?loop systems can be obtained in Fig. 4. It is observed that there exist a resonance frequency and an anti?resonance frequency in the humanoid robot control system. As to amplitude characteristics,the open?loop gain at resonance frequency in?creases sharply, and the case at the anti?reson?ance frequency is the opposite. The closed?loop pole is farther away from the imaginary axis than the open?loop pole, that is, the closed?loop reson?ance frequency is larger than the open?loop res?onance frequency. Therefore, the amplitude char?acteristics vary dramatically at the frequency equal to NTF and ARF as a consequence of joint flexibility. It is not beneficial to the stable opera?tion of the system.

Fig. 4 Bode plots of the original system

The vibration caused by imaginary conjug?ate poles belongs to free vibration. In other words, if anti?resonance occurs, the motor speed keeps stable, and load speed oscillates, which has a trend of attenuation. It will not be excited by external disturbances. The vibration caused by imaginary conjugate zeros belongs to self?excited vibration. If control frequency equals the reson?ance frequency, the system will oscillate sharply and will not show a convergence trend.

Because of the inherent flexibility of the joint, the vibration caused by anti?resonance fre?quency appears as the residual vibration of the system. The vibration caused by resonance fre?quency appears as the resonance phenomenon ex?cited by control frequency.

The time domain response and the frequency domain response of the system at the resonant frequency are obtained, as shown in Fig. 5 and Fig. 6. Fig. 5 illustrates the motor speed response of the closed?loop system at the resonant fre?quency. It can be observed that the amplitude seriously exceeds the given amplitude and the motor speed oscillates. The motor speed appears serious overshoot and the speed response appears high frequency oscillation in Fig. 6.

Fig. 5 Unit sinusoidal response of closed?loop system

Fig. 6 Step response of motor speed

Based on the analysis mentioned above, if control frequency equals to the resonance fre?quency of the humanoid robot arm, the reson?ance will occur. Due to the coupling relationship between arm and motor, when the arm resonates,on the one hand, the speed and torque oscillate in the mechanical system. On the other hand, the output current and voltage amplitude fluctuate in the electrical control system. In the following,a PR controller is designed to deal with the above problem.

2 Resonance Suppression of Humanoid Robot Arm

2.1 PR controller

The humanoid robot arm resonates mainly at the specific frequency. If the control system frequency reaches the frequency at the resonance frequency, the arm will resonate. In order to sup?press the resonant phenomenon and improve sys?tem stability, a PR controller is introduced to the humanoid robot control system.

where Kpand Kiare the proportional and the in?tegrator coefficients, respectively. Take Eq. (12)into Eq. (11), we have

Fig. 7 Bode plot of open?loop PR current control system

Fig. 8 shows the frequency response wave?forms of a PR controller when the cutoff frequen?cies are 5 rad/s, 10 rad/s, and 15 rad/s, respect?ively. The gain at the given resonant frequency changes from infinite to a finite value Kp+Kr.Compared with the ideal PR controller, the cutoff frequency ωccan be set appropriately to expand the controller bandwidth so as to de?crease the sensitivity to the signal frequency vari?ation, which improves the stability of control sys?tem. In practice, ωcwith the value of 5–15 rad/s is appropriate to ensure a suitable bandwidth[17].

Fig. 8 Bode plot of open?loop PR current control system with different ωc

2.2 Resonance suppression for humanoid robot arm based on PR controller

In the driving control system of humanoid robot, the control strategy of drive system for humanoid robot arm joint is designed based on the dynamic model of robot arm. The structural diagram of resonance suppression for humanoid robot arm based on the PR controller is shown in Fig. 9.

Fig. 9 Structural diagram of resonance suppression for hu?manoid robot arm

When control frequency equals the resonant frequency of arm, the resonance phenomenon oc?curs. According to the dynamical equation of the humanoid robot arm, there is a coupling relation?ship between the motion state of the arm and motor. The driving motor voltage will fluctuate when the motion state of the arm changes, which lead to output fluctuations of the controller un?der the circumstance of given input current. In the robot arm controller, the voltage fluctuation is given as a disturbance of the current closed loop.

The current?disturbance ratio rejection cap?ability at null reference, for the control structure,demonstrated in Fig. 10, is defined as

Fig. 10 Block diagram of the current loop based on PR con?troller

The Bode plot of disturbance rejection for a PR controller is shown in Fig. 11. As it can be observed, around the resonance frequency the PR controller provides a large attenuation of amp?litude. Besides, it is clear that PR has the char?acteristic of rejection capability at a specific fre?quency from Fig. 8.

Fig. 11 Bode plot of disturbance rejection of a PR controller

Fig. 12 Root?locus of the current controller

Fig. 12 illustrates the eigenvalue of the con?trol system distribute in left semi?plane on s do?main under different time constants. When the damping ratio of the control system is between 0.4 and 0.8, the system overshoot is small, and the speed response is fast. The expected damp?ing performance can be obtained through the root?locus method. The range of the proportional coefficient is Kp∈[0.11,0.75], and the range of the resonant coefficient is Kr∈[37,250].

A PR controller is introduced to the hu?manoid robot arm system and its the block dia?gram is shown in Fig. 13. Fig. 14 shows the Bode plot of the open?loop and closed?loop system of robot arm with the PR controller. The gain of open?loop system attenuates to 0.203 dB at the resonant frequency. The PR controller adds damping to the system and increases gain mar?gins of the system. It is observed that the gain of the open?loop resonance frequency of the system can be effectively attenuated after the PR con?troller is added to the current loop. Additionally,the step response of the system using the PR controller is shown in Fig. 15. It can be seen that the oscillation is eliminated and the response speed increases. The system stability is improved.

Fig. 13 Block diagram of humanoid robot arm with PR con?troller

Fig. 14 Bode plot of open?loop and closed?loop system with PR controller

Fig. 15 Step response of motor speed with PR controller

3 Simulation and Analysis

In order to verify the effectiveness of the res?onance suppression scheme for the humanoid ro?bot arm based on PR controller, this paper uses MATLAB/Simulink to build a model for simula?tion verification.

The parameters of the BLDC motor are as follows: pole pairs p=4, the rated voltage is 12 V,the armature resistance R is 24 Ω, the rated tor?que is 0.5 N·m, the rated speed is 1 000 r/min,the equivalent inductance is 1.4 mH, and the ro?tational inertia coefficient is 0.000 62 kg·m2. The moment of inertia of the humanoid robot arm is 0.004 5 kg·m2, and the link length is 0.15 m.The joint stiffness coefficient is 10.

Fig. 16 Simulation waveforms without PR controller

The disturbance signal with the frequency equals resonance frequency is added in simula?tion. The results imply that the voltage disturb?ance causes an additional and unexpected cur?rent, resulting in current distortion in three?phases. Besides, it is observed that the rotation?al speed and torque oscillates.

Simulation results of phase current, speed and torque waveforms when utilizing the PR con?troller are illustrated in Fig. 17. As shown in Fig. 17a, when the resonance occurs, under the control of the proposed PR controller, the cur?rent waveforms are close to an ideal trapezoidal form at 0.15 s. The current distortion is reduced.Fig. 17b shows the A?phase current waveforms.After 0.15 s, the motor output torque remains stable with a little fluctuation. At 0.15 s, the speed reaches 1 000 r/min, which is within the al?lowable error range. And then the speed keeps stable.

Compared with the simulation waveforms under voltage disturbance without the PR con?troller, it can be seen that PR controller has the good effect of disturbance rejection at the specif?ic frequency. When the disturbance is suppressed,the phase current maintains rectangular wave?forms, the speed curve is stable, and the torque has little fluctuations. It is obvious that the res?onance suppression for humanoid robot arm based on PR controller is effective. As a result,the dynamic performance and system stability are improved.

Compared with the simulation waveforms under voltage disturbance without the PR con?troller, it can be seen that PR controller has the good effect of disturbance rejection at the specif?ic frequency. When the disturbance is suppressed,the phase current maintains rectangular wave?forms, the speed curve is stable, and the torque has little fluctuations. It is obvious that the res?onance suppression for humanoid robot arm based on the PR controller can effectively sup?press the resonance phenomenon. As a result, the dynamic performance and system stability are improved.

Fig. 17 Simulation waveforms based on PR controller

4 Conclusion

This paper presents a resonance suppression strategy for a humanoid robot arm based on a PR controller, which solves the problem caused by the control frequency and resonance frequency,since the driving motor voltage will fluctuate when the motion state of the arm changes. The fluctuation is regarded as the disturbance of closed?loop current control. The PR controller has the characteristic of disturbance rejection at the specific frequency without affecting the per?formance at other frequencies. The output cur?rent of the inverter will not be affected by the vi?bration of the arm at the resonance frequency.Consequently, the control frequency will not equal to the arm resonant frequency so that the self?excited oscillation will not be excited. The simulation results show that the proposed stra?tegy has a good resonance suppression effect at a specific frequency. The total system has a very simple structure and the resonance of humanoid robot arm was well suppressed by the proposed method. The new strategy also improves the sta?bility of the system, ensuring that the system has a good dynamic performance.