運動控制概論

題目： Torque Sensorless Control in Multidegree-of-Freedom Manipulator

作者： Toshiyuki Murakami, Fangming Yu, and Kouhei Ohnishi出處： IEEE TRANSACTIONS ON INDUSTRIAL

ELECTRONICS, VOL. 40, NO. 2, APRIL 1993 P259~P265

報告人：控晶四乙─黃筌園指導老師：王明賢 老師

Abstract-The paper describes a torque sensorless control in amultidegree-of-freedom manipulator. In the proposed method,two disturbance observers are applied to each joint. One is used to realize the robust motion controller. The other is used to obtain a sensorless torque controller. First, a robust acceleration controller based on the disturbance observer is shown. To obtain the sensorless torque control, it is necessary to calculate reaction torque when the mechanical system performs a force task. Second, we explain the calculation method of the reaction torque. Then the proposed method is expanded to workspace force control in the multidegree-of-freedom manipulator. Finally,several experimental results are shown to confirm thevalidity of the proposed sensorless force controller.

摘要

I. INTRODUCTIONII. OBSERVER-BASMEDO TIONC ONTROLLERIII. SENSORLESS FORCECO NTROLLER IN

WORKSPACEIV. EXPERIMEV. CONCLUSIONSVI. REFERENC

I.INTRODUCTION The paper deals with an advanced torque control technique

to realize sensorless torque control in the multidegree-of-freedom manipulator. If the generated torque is known, it will bring some sophisticated abilities to the manipulator. For example, impedance control is one of the important applications. From the viewpoint of motion control, any motion control is attained precisely if the generated torque is well controlled. To realize the precise torque control, it is necessary to detect the transient generated torque by more than one torque sensor. This makes the structure of the drive system complicated. In addition, the sensor output is affected by unknown disturbances such as temperature variation. Since torque sensor is not convenient from a mechanical and an economic viewpoint, it is strongly desirable to realize the torque control without the torque sensor [6].

In this paper, a sensorless torque control is achieved byapplying a disturbance observer at each joint, thus estimatingthe disturbance torques imposed on the manipulator [l], [3], [4]. First, the estimated disturbance torque is directly fed back to realize the acceleration controller whose input is the acceleration reference. The motion system based on the acceleration controller generates the desired acceleration without any disturbance effect. Second, the reaction torque is calculated from the estimated disturbance torque and the identified dynamic model of manipulator. By the feedback of the calculated reaction torque, the sensorless torque control is realized. The proposed algorithm is very simple and easy to apply to the control system. In this paper, the algorithm is extended to the workspace force control of a multidegree-of-freedom manipulator.

The paper consists of five sections. In Section 11, thedisturbance observer is explained. The acceleration controlleris also introduced. Sections I11 and IV present the sensorless force controller in the multidegree-of-freedom manipulator and several experimental results. The conclusions are summarized in Section V.

II. OBSERVER-BASMEDO TIONC ONTROLL

A. Robust Motion Controller Based on Disturbance Observer Fig. 1 shows a model of an electrical motor and manipulator.Here, Ktni and Jni are the nominal torque coefficient and inertia,respectively. The system reference is the torque current Reference . Tdisi is the total disturbance torque imposed on the system, which is defined in (1). Here, the lower suffix i denotes the joint number in the multidegree-of-freedom Manipulator.

On the right-hand side of (0, the first term is the interactive torque, including the coupling inertia torque, gravitation, and so on. The second term is the reaction torque when the mechanical system does force task. The third and fourth terms are the viscosity and Coulomb friction,respectively. The fifth term is the self-inertia variation torque. The last term models the torque pulsation due to the flux distribution variation of the motor. As shown in (l), Tdisi

includes not only the general load torque, but also the parameter variations. From Fig. 1 and (l), we obtain (2a), which is the joint motion equation of themanipulator:

Equation (2b) means that the disturbance torque Tdisi canbe calculated from the angular acceleration of motor qiand the current reference In general, it is difficult todetect the angular acceleration directly by an industrialgradesensor. So we obtain the acceleration by differentiatingthe angular velocity. Then, a low-pass (LP) filter is inserted to reduce the noise included in the velocity signal.

The total calculation process of the disturbance torque is shown in (3a) and Fig. 2(a). In general, it is difficult to implement the ideal differentiator, so a pseudodifferentiator is used to obtain the angular acceleration as shown in the second part of (3a). In Fig. 2(a), the torque is the sum of and Icmpi, which is thecompensated current to cancel out the disturbance effect.Then the total system becomes robust against the disturbanceeffect including the parameter variations:

In the disturbance observer shown in Fig. 2(a), the filter design relates to the approximation of the disturbance effect. When the filter element is first order, the disturbance effect is approximated by the step change function,as shown in Fig. 3. Therefore, the disturbance effect is estimated more precisely if the order of the filter is higher. Then, the higher the order is, the more the calculation of the observer becomes complicated. In case the sampling ratio of the control system is small and the time delay of the filter is negligibly small, the disturbanceeffect is well estimated by the first-order system. So we use the first-order filter to simplify the calculation of the disturbance observer. In Fig. 3, we assume a discrete-time system. In the real implementation, the sampling ratio is negligibly small (1 ms), so that we introduce a continuous-time discussion in the filter representation and the latter analyses.

Fig. 2(a) is transformed into the hatched block in Fig.2(b) by using (3b). Here, Gsi(s) is the sensitivity function of the system against the disturbance torque. The smaller Gsi(s) is, the more robust the system becomes. That is to say, the robustness of the total system is evaluated by the sensitivity function Gsi(s) [l], [3], [8]. In the observer-basedsystem, the robustness and the robust stability of the system relate to the selection of g. The larger g is, the more robust the system becomes. On the other hand, the smaller g is, the more the stability margin of the system increases. This means that there exists a tradeoff in the selection of g. In general, it is possible to select g large enough so that the disturbance effect is well suppressed, that is, Gsi(s) = 0. When the gain Jni/Krni is inserted to determine the current reference , Fig. 2(b) is theacceleration controller whose input is the accelerationreference. Then the desired acceleration is generated without any disturbance.

In the manipulator based on this joint acceleration controller, high-performance motion control is expected by the feedback of the state variables of the manipulator.In force control applications, it is necessary to feed back the reaction force in the outer loop. As mentioned before, the observer-based manipulator is robust against the disturbance torque, including the reaction torque, so that the manipulator will apply the infinite force to the environment which the manipulator contacts in case the reaction force is not fed back. From this point of view, the force sensor is essential to realize the force control in the observer-based system. Of course, in the conventional force controller, the force feedback is important to obtain the high-performance force control. As described previously,this makes the structure of the manipulator system complicated. To improve this matter, the paper proposes the force control strategy without using the force sensor.In the next section, the calculation method for the reaction torque is shown.

B. Calculation of Reaction Torque

In this section, we explain the calculation method of the reaction torque by using the output of the disturbance imposed on the manipulator is defined in (1) and is estimated by the disturbance observer. If the total disturbance effect is known, the reaction torque can be calculated as follows:

Here, we assume that the nominal values Jni and Ktni are well adjusted, that is, these values are almost equal to the real values Ji, Kti. In fact, it is possible to know the values of Ji and Kti in advance by implementing several motion tests. Under this assumption, (4a) is rewritten as follows:

ln (4b), Tinti is the interactive torque imposed on each joint. Here,we introduce, the serial multilink manipulator, which has one electrical motor at each joint. In this case,the theoretical equation of Tinti is derived from the Euler-Lagrange equation. Using the Euler-Lagrange equation, the end-effector equations of motion are givenbY

where t(q,q) and u(q) are the kinetic energy and the potential energy respectively. is the joint torque and I (q) is the joint inertia matrix. We assume that is the sum of the input torque Kt , the reaction torque Text, and the friction effect f(q). From (5), we obtain (6), which are the motion equations of the manipulator. Here,Kt is a diagonal matrix whose elements are the torque coefficients of each joint:

In (6), the joint inertia matrix z ( q ) changes according to the configuration of the manipulator, and the matrix of torque coefficient Kt changes due to the variation of flux distribution of the rotor. These effects are represented by (7), and it is substituted into (6). As a result, we obtain (8), which is the vector formulation of (1) and (2). Here, Ktn and In are the diagonal matrices whose (i, i ) elements are the nominal torque coefficient and the nominal inertia of the ith joint, respectively.

From (1) and (8), Tnf is defined as follows:

• Using (4) and (9), the reaction torque Text is calculated.Here, Tint and f(4) are the inverse dynamics swith respect to the disturbance torque. The total CalCUlation process of the reaction torque is summarized in Fig. 4(a). In the actual application, (3a) is expanded into the calculation of the reaction torque.olution.

Then the calculation ofFig. 4(a) is Performed as follOws:

• Equation (10) shows the observer structure of each joint to calculate the reaction torque. In this paper, the observer shown in (10) is called the torque estimator. In this calculation process, it is necessary and important to know the gravity and the friction effect as precisely as possible.In fact, they are the main disturbance effects and have highly nonlinear characteristics. Excepting the gravity term, all parameters of (9) are calculated from the motion

• equations of the manipulator. Fortunately, in the observer-based system, the gravity and the friction effects are measured precisely by implementing the constant angular velocity motion tests. In the next section, the measurement method of the gravity and the friction effects is shown.

C. Measurement of Gravity and Friction EffectThe gravity and the friction effects can be measured inadvance by using the estimated disturbance torque of thedisturbance observer. Under the constant angular velocitymotion test, the estimated disturbance torque is given by

In the two-link manipulator of Fig. 5(a), which uses joint2and joint3, (11) is rewritten as follows:

All parameters A2, A3, f2(q2), f 3 ( q 3 ) in (12) with respect to each velocity are determined by repeating the constant angular velocity motion tests. The calculation of Fig. 4(a) is performed by using these measured parameters.

III. SENSORLESS FORCECO NTROLLER IN WORKSPACE

As mentioned in the former part, the robust accelerationcontroller is realized by the feedback of the estimateddisturbance torque. In addition, we can obtain thereaction torque from the torque estimator shown in (10).In this part, the sensorless force controller is constructedbased on the joint acceleration controller and the torqueestimator in the workspace.

IV. EXPERIMEIn this part, several experimental results are shown to confirm the validity of

the proposed sensorless force controller.A. Experimental SystemFig. 5(a) shows the tested direct drive manipulator which has three degrees-Of-freedom. In this system, one direct drive motor is attached to each joint, and the force sensor is mounted at the tip of manipulator. The force sensor is used to compare the estimated reaction force with the real reaction force. The specifications of the DD motor are summarized in Table I. The signal flow Diagram of the control system is illustrated in Fig. 5(b). The sampling period of the controller is 1 ms, and all calculations are performed by DSP (digital signal processor: pPD77230). In the experiments, joint2 and joint3 are used toimplement the workspace force control. First, the results of the constant angular velocity motion test are done to measure the gravity and the friction effects. Second,the sensorless force control based on the proposed methodare implemented by the DD manipulator.

B. Constant Angular Velocity Motion TestTo measure the friction and the gravity effects, two observers are constructed in each joint. One is used to realize the robust motion controller. The other is used to measure the friction and the gravity effects. Especially the latter observer is called the dynamics identification observer. First, to measure the gravity effect, the constant angular velocity motion tests are implemented. In this case, the dynamics identification observer has the same structure of the general disturbance observer. The tested motions are illustrated in Fig. 6(a). In Fig. 6(a), joint3 moves from 0 [radl to 27r [radl with constant velocity, and joint2 is fixed to 0 [rad]. Then we obtain the estimated disturbance torque of joint3 as shown in Fig. 7(a). From this result, we can find it easy to know the gravity effect represented by the sinusoidal function.

Then, its amplitude corresponds to A, in (12). On the other hand, in Fig.6(b), joint2 moves from 0 [radl to 27 [rad] with constant velocity, and joint3 is fixed to 0 [rad]. Fig. 7(b) is the estimated disturbance torque of joint2. In this result, the amplitude of the sinusoidal wave is A, + A,. From these results, A, and A, are calculated separately. The friction effect of each joint is also measured by the constant angular velocity motion test. Then, the dynamics identification observer is constructed as follows:

In the calculation of (16), only the friction effect is estimated.As described before, the estimated disturbance torque of the disturbance observer is fed back to cancel out the total disturbance effect imposed on the manipulator.So the velocity control is well realized, and the friction effect is measured in the wide range of velocity. This is a remarkable point of the proposed identification method. In the conventional method, it is difficult to know the friction effect in the low-velocity area. The measured friction effects are hown in Fig. 8. The results obtained by the constant angular velocity motion tests are summarized in Tables I1 and 111. To calculate the reaction torque, the obtained results are used.

C. Sensorless Force ControlFig. 5(a) shows the tested motion of sensorless force control. The force command is applied to the -y direction and the position control is done in the +x direction.Fig. 9 is the step response of sensorless force control. In the upper result of Fig. 9, the force response coincides with force command. This means that the disturbance torque is well cancelled out by the feedback of the estimated disturbance torque, and the high-performance forcecontrol is realized. The lower result of Fig. 9 shows that the calculated reaction force and the output of force sensor are almost same. Fig. 10 shows the correlation between the calculated reaction force and the output of force sensor in case the sinusoidal force reference is applied to the sensorless force controller. From these results, we can find that the reaction force is well estimated by the proposed method. These results also show the feasibility of the proposed sensorless force control.

V. CONCLUSIONSThe paper presents a novel approach to sensorlesstorque control. This is one of the industrial applications ofthe disturbance observer. In the observer-based manipulator,the following abilities are added to the manipulator. ●The robustness of the manipulator is increased by the feedback of the estimated disturbance torque.●By resolving the estimated disturbance torque, it ispossible to know each nonlinear effect imposed on the manipulator and to obtain the exact model of manipulator.

In the DaDer, the sensorless torque control is realized by using ;he above abilities. The fiist ability is important to realize the robust force controller. In the proposed sensorless force control, it is necessary to know the friction and the gravity effects as precisely as possible since these effects give much effect to the force response.Under the constant angular velocity motion test, these effects are measured by using the estimated disturbance torque, which is the application of the second ability. The feasibility of the proposed sensorless force control is confirmed by several experiments.

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