Tracking Control Of An Electrohydraulic Manipulator In The Presence ...
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Tracking Control of an Electrohydraulic Manipulator in the Presence of Friction Shahram Tafazoli, Member, IEEE, Clarence W. de Silva, Fellow, IEEE, and Peter D. Lawrence, Member, IEEE
Abstract—Analysis and estimation of friction and compensation for its effects in the control of an electrohydraulic manipulator is addressed in this paper. The specific hydraulic manipulator is an integral part of an automated fish processing machine which has been developed in our laboratory. The analysis reveals that considerable static and dynamic friction exists in the system. An available nonlinear observer for Coulomb friction, is modified to simultaneously estimate friction, velocity, and acceleration. A novel observer-based friction compensating control strategy is developed for improved tracking performance of the manipulator. The approach is based on acceleration feedback control. Experimental investigations show that this controller significantly outperforms a conventional, proportional-plus-derivative (PD) controller. The general approach presented in this paper, may be applied to compensate for friction in any servomechanism, particularly when the actuator dynamics is not negligible. Index Terms—Acceleration control, error analysis, mechanical factors, observers, parameter estimation, servosystems, tracking.
I. INTRODUCTION
A
N industrial automation laboratory has been established primarily for research and development of advanced technology in automation of the fish processing industry [1]. In this laboratory, an automated machine for mechanical processing of salmon has been developed. In this industrial prototype, the fish are transported using a conveyor system. When a fish reaches the cutter blade, its image is taken by a digital camera. A computer analyzes the image and ) for removing generates the desired position coordinates ( the head, with minimal wastage of meat. The control system of two independent ( and ) electrohydraulic actuators then performs precise positioning of the cutter blade assembly. In order to meet the process requirements, this motion has to be fast, accurate, smooth, and nonoscillatory. After the blade is moved to the desired position in the horizontal plane, a pneumatic actuator that moves the blade in the vertical ( ) direction is activated, in order to remove the fish head. Note that the cutter positioning mechanism is in fact a Cartesian
Manuscript received November 3, 1995. Recommended by Associate Editor, F. Svaricek. C. W. de Silva was supported by the NSERC Industrial Automation Chair Grant 5-80595 and B.C. Advanced Systems Institute Fellowship Grant 5-56296. P. D. Lawrence was supported by NSERC NCE IRIS Grant 5-58305. Additional funding was provided by B.C. Packers, Ltd. and Garfield Weston Foundation. S. Tafazoli and P. D. Lawrence are with the Electrical and Computer Engineering Department, University of British Columbia, Vancouver, BC, Canada V6T 1Z4. C. W. de Silva is with the Mechanical Engineering Department, University of British Columbia, Vancouver, BC, Canada V6T 1Z4. Publisher Item Identifier S 1063-6536(98)03090-5.
electrohydraulic manipulator, and is an integral part of the fish processing machine. The sliding surfaces in the two axes generate frictional forces. Since friction can degrade the system performance, it is undertaken to identify it, with the objective of compensation for its effects in closed-loop control. Analysis and on-line estimation of the frictional force in the system has been studied in our previous work [2]. Therein, we found that a considerable amount of static and dynamic friction exists in the prototype system. The frictional force arises primarily due to the greaselubricated metal guideways on which the blade assembly slides, and also due to rubbing of the hydraulic pistons against their cylinder walls. In [2], a nonlinear observer was developed that simultaneously estimates the dynamic friction and velocity. Position and force control of the electrohydraulic actuator was studied subsequently, in [3]. In this paper, the previous investigations are combined and enhanced in order to gain a thorough insight into both the behavior of friction in the prototype system and the performance of the friction-compensating controller. Furthermore, a practical method is developed for choosing the gains of the controller and the associated observer. Even though a specific machine (an electrohydraulic servomechanism) is studied here, the overall control strategy that is developed is quite general and can be applied to any mechanical actuation system, especially when the actuator dynamics is not negligible. Friction is a complex, natural phenomenon which is present in virtually all mechanical control systems including electric, pneumatic, and hydraulic actuators. It can pose a serious challenge to achieving high-quality performance of these machines. In particular, it plays an important albeit damaging role in hydraulic control systems [4]. Friction can lead to tracking errors, limit cycle oscillations, and undesirable stick-slip motion. As a result, modeling of friction and compensation for its effects has received considerable attention from researchers [5]–[8]. In the present paper, we will particularly focus on the observer-based friction estimation and compensation technique that has been proposed by Friedland et al. [8]–[11]. Acceleration-assisted tracking control has been discussed by de Jager [12], where a performance improvement by a factor of up to 1.5 is reported in the particular application. According to de Jager, acceleration measurement in mechanical control systems can be used to decrease the tracking error, improve the robustness for modeling errors, and reduce the effective friction. Studenny et al. [13] have successfully applied acceleration feedback control to robotic manipulators. An alternative method for friction compensation is to measure the transmitted
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TABLE I PHYSICAL PARAMETERS OF THE PROTOTYPE MANIPULATOR
Fig. 1. The high-speed electrohydraulic positioning system (for one axis).
force (torque) and close a force (torque) feedback loop. The approach, termed joint torque sensory feedback, has been applied for force control of robotic manipulators [14]–[16]. In the system that is considered in the present paper, there is no sensor to directly measure the acceleration or the transmitted force. Instead of direct measurement, acceleration will be computed from the estimated friction. Then, by combining friction estimation and acceleration feedback control, an adaptive friction compensation technique is established. This paper is organized into several sections in the following manner: In Section II, the experimental system is briefly described. In Section III, a general state-space model is derived for the electrohydraulic manipulator, primarily to show the nonlinear nature of the actuator dynamics. In Section IV, the nonlinear observer for friction parameter, proposed by Friedland and Park [8], is discussed and its convergence rate is analyzed. In this section, a new combined observer is developed for simultaneous estimation of friction, velocity, and acceleration. In Section V, the experimental results on friction analysis and estimation are presented. In Section VI, a new friction-compensating control strategy is introduced which is based on acceleration feedback control. In Section VII, performance of the proposed controller is experimentally investigated with comparison to that of a conventional proportionalplus-derivative (PD) controller. Conclusions are outlined in Section VIII. II. THE EXPERIMENTAL SYSTEM The prototype electrohydraulic manipulator is of Cartesian type with two orthogonal prismatic joints. It has two control inputs which are the currents applied to the electrohydraulic proportional valves for the and directions. Since the two degrees of freedom in the manipulator are quite similar and also independent with regard to operation, only the direction is studied in this paper. Fig. 1 shows the simplified schematic diagram of the manipulator, for the axis. The servovalve consists of a pilot stage with torque-motoractuated flapper and a spring-centered boost stage with double sliding spool arrangement. The differential pressure from the
pilot is applied across both spools of the boost stage and is balanced by the centering springs, producing a spool position that is proportional to the differential pressure and therefore to the input current. The nominal full flow current of this mA. two-stage servovalve is A Temposonics linear displacement transducer is installed at the head of the hydraulic cylinder. This industrial magnetostrictive sensor precisely senses the position of an external magnet, which is connected to the piston, from the time interval between an interrogation pulse and a return pulse. The specific sensor that is used has a resolution of 0.025 mm and its measurement update time is less than 1 ms. Two gage-pressure transducers are also installed on the head and rod sides of and the cylinder, in order to measure the fluid pressures , respectively. Note that there is no velocity or acceleration sensor in the system. All the sensor outputs (including pressure transducers) are low-pass filtered by first-order antialiasing kHz is used RC filters. A sampling frequency of to implement the digital controllers; accordingly, the cutoff frequency of the filters is set at 200 Hz. A 12–bit PC-based data acquisition system is used for data collection and graphical display. The programs have been written in C++. The physical parameter values of the prototype manipulator are listed in Table I. III. MODELING OF THE ELECTROHYDRAULIC SYSTEM A dynamic model is derived in this section for the electrohydraulic manipulator. This analysis is intended primarily to emphasize the nonlinear nature of the actuator dynamics. Similar approaches to modeling of asymmetric hydraulic actuators have been reported in [17] and [18]. Primarily, there are three types of nonlinearities in the servovalve system: namely, the basic flow equation through an orifice, the hysteresis resulting from electromagnetic characteristics of the torque motor, and the flow forces on the valve spools [4], [17]. Displacement of the valve spools from their null position creates a pressure difference across the single-rod hydraulic actuator and the resulting fluid flow causes a change in the position of the piston. The force applied by the actuator can be calculated from cylinder pressures. Newton’s second law applied to the actuator yields (1) where force,
is the actuator force, is the opposing frictional is the piston displacement, and is the equivalent
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Fig. 2. A simplified block diagram of the system.
moving mass in the direction. The amount of fluid flow to and from the rod-side of the cylinder the head-side is a function of both the valve spool position and cylinder pressures. Assuming that the valve is of critical-center type with matched and symmetrical ports (see [4] for definitions), the relationship can be expressed in the following form [18]:
Fig. 3. System model with Coulomb friction.
required in the control strategy that is proposed in the present paper. IV. ESTIMATION OF FRICTION, VELOCITY, AND ACCELERATION
(2) A. The Friedland–Park Friction Observer [8] is the valve spool displacement, is a fixed gain, where is the supply pressure, and is a switching function defined as below (3) . Furthermore, using the continuity principle, and taking fluid compressibility into account, one can write [4] (4) is where is the effective bulk modulus of the fluid and the volume of the fluid inside each of the hoses that connect the servovalve to the actuator. Next, define a state vector for the system as follows: (5)
In a dynamic experiment, an observer is needed to estimate the frictional force. Since the force applied by the actuator can be easily calculated from pressure readings, the nonlinear observer proposed by Friedland and Park in [8] for Coulomb friction, appears to be suitable in the present application. The Friedland–Park observer is briefly discussed in the following. Note that their approach is reformulated here in order to include the object mass . System dynamics is assumed to be as shown by the block diagram in Fig. 3. In this figure, the block denoted by “NL” represents the nonlinear friction element. The frictional force is assumed to be of classical Coulomb type, as given by (7) To estimate the Coulomb friction parameter , Friedland and Park proposed the following nonlinear observer:
By combining (1), (2), and (4), one arrives at the following nonlinear state-space model of the system:
(8) is the estimated Coulomb friction parameter, where denotes the state variable of the observer, and the design and the exponent . The parameters are the gain variable represents the estimated acceleration and is given by
(6) (9) where
is the estimated Coulomb friction; i.e., sgn
A simplified block diagram of the hydraulic manipulator is shown in Fig. 2. In this diagram, the “actuator dynamics” block represents the nonlinear hydraulic actuation system. Servovalve dynamics can also be included in this block. According to the state-space equations, the actuator dynamics depends on the position and velocity of the piston. Vossoughi and Donath [17] have discussed various nonlinearities that are present in electrohydraulic systems. They used feedback linearization to compensate for these effects in the control design. Knowledge of the manipulator dynamics is not
(10)
Fig. 4 shows a block diagram of the Friedland–Park nonlinear observer. Defining estimation error as , where is the true friction parameter, Friedland and Park have shown that the error dynamics is given by (11) Therefore, for a fixed true parameter ( ), the estimation error converges asymptotically to zero if and is bounded away from zero. By simulation, Friedland and Park
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Fig. 5. The velocity filter.
Fig. 4. The Friedland–Park Coulomb friction observer.
have also confirmed that the observer works well for more is a nonlinear function complicated friction models where of velocity. B. Convergence Rate of the Friedland–Park Observer Convergence rate of the Coulomb friction observer that has been proposed by Friedland and Park, is analyzed now. This is an original contribution of the present paper. According to (11), the estimation error has linear, time-varying dynamics. ), explicit solution of Assuming a fixed true parameter ( , gives (11) for an initial estimation error (12)
As discussed in [2], the observer proposed by Friedland and Menzelopoulou must be modified in order to accommodate the static friction component in our system. In the modified observer, velocity estimation is performed independently of the friction estimation. The velocity observer is in fact a firstorder low-pass filtered differentiator, and is represented by the is the observer transfer function given in Fig. 5, where gain. The estimated velocity is then used in the Friedland–Park algorithm. Implementation of the modified observer for friction estimation and compensation in a dc-motor-driven apparatus has been recently reported in [11], where the authors use the name “Tafazoli modification” for the proposed observer. Their experimental investigations show that the performance of the observer is very promising, such that friction estimates obtained compare favorably to the actually measured friction, and significant tracking improvement is obtained by compensating for the estimated friction in the control law. Since the estimation update is poor at low velocities, the observer estimates dynamic friction only. Nevertheless, static friction has to be taken into account for estimating the net friction. The following interpolating function will be used for this purpose: (15)
Thus, nonzero velocity drives the estimation error to zero. Equation (12) can be written in the following form, to establish an analogy with first-order linear time-invariant observers
with (13) It follows that, the approximate pole of the observer is located at (14) denotes the mean value of the corresponding variwhere able. Equation (14) can be used to choose the gain and the exponent for a desired rate of convergence. C. Simultaneous Estimation of Friction, Velocity, and Acceleration In [9], Friedland and Mentzelopoulou developed a combined observer for friction and velocity in the absence of direct velocity measurement in the system. The authors employed an observer for velocity estimation and then used this estimate in the Friedland–Park algorithm. They showed local asymptotic stability for both velocity and friction estimations.
This empirical function is adopted from [5] and [6], where it has been used in a different context. In (15), is the applied force, is the net friction, is the Coulomb (dynamic) friction, and is the threshold velocity. A small neighborhood of zero velocity is defined by , as in the model proposed by Karnopp [20]. According to (15), outside this neighborhood, friction is mainly dynamic in nature (a function of velocity). Inside the neighborhood, velocity is considered negligible, and static friction is dominant (a function of the applied force). This formulation is used to account for noise and numerical inaccuracy of the estimated velocity. Thus, the net frictional force can be computed from the following equation: (16) Likewise, the estimated acceleration would be (17) The variable in (9) represents the acceleration signal provided that the velocity is away from zero. By combining (9) and (17), we obtain (18) Thus, the estimated acceleration is an enhanced version of , as static friction is taken into account in its calculation. Fig. 6
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Fig. 6. The combined observer. Fig. 7. The quasistatic sinusoidal response of the open-loop system.
shows the final form of our new observer for velocity, friction, and acceleration. The design parameters for the observer are , , , and . V. ANALYSIS AND ESTIMATION OF FRICTION: EXPERIMENTAL RESULTS In this section, the characteristics of friction at very low velocities is investigated. Also, off-line and on-line estimation of friction and velocity in the prototype manipulator, are addressed. The Euler approximation is used for digital implementation of the nonlinear observers.
Fig. 8. Friction-velocity characteristic curve at low velocities.
A. Friction-Velocity Characteristic at Very Low Velocities In a quasistatic experiment, when the piston moves very slowly, the inertial force in (1) can be neglected compared to . Then, the applied force is a good representation of the frictional force. A sinusoidal current with an amplitude of 2.5 mA (7% of the rated current) and a period of 100 s was applied to the servovalve. The pressures and the piston position were sampled every 2 s. Note that in this system, piston movement is quite limited. In view of this, the applied current to the servovalve, in open-loop experiments, should be small. To estimate velocity from position signal, a digital filter was employed off-line, which is in fact a high-order, delay-free, low-pass-filtered differentiator [2]. Fig. 7 shows the result of this test. According to this figure, there is a significant amount of static friction that causes deadband nonlinearity in the open-loop system [4]. The effect of this nonlinearity on the closed-loop response will be explained later. The measured force and the estimated velocity were used to obtain the steady-state, friction-velocity characteristic, and is plotted in Fig. 8. Note that only one response cycle is used to obtain this curve. The curve was found to be slightly
different for different cycles, which is due to the random characteristics of friction. It is evident from Fig. 8 that the friction behavior resembles that of classical Coulomb friction with some hysteresis effect. However, the transition from static friction to a dynamic friction regime is not abrupt, contrary to the classical friction model [5]–[7]. Finally, acceleration was estimated by applying a similar digital filter to the velocity signal (to obtain ). As shown in is quite negligible compared to Fig. 9, the inertial force the applied force . B. Off-Line Estimation of Friction and Velocity In a dynamic experiment, the Friedland–Park nonlinear observer can be used off-line, employing the digital differentiator discussed in Section V-A to estimate velocity. A digital proportional controller (with a gain of mA/m) was used to close the feedback loop, at a sampling frequency of kHz. The desired position trajectory was chosen as m
(19)
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Fig. 9. The estimated inertial force in the quasistatic experiment.
Fig. 11. Estimated dynamic friction as a function of velocity.
Fig. 10. Off-line estimation of Coulomb friction using the Friedland–Park observer ( = 1:5; ka = 150).
which gives the following expression for the desired velocity: m/s
(20)
To study the bandwidth of the observer, assuming that the tracking error is not very large, one can replace the actual in (14) with the desired velocity velocity (21) , in order to obtain a high speed of convergence, Since . it is desirable to choose the exponent in the range , the expression for the observer pole becomes With
the Friedland–Park observer (with ) to the recorded velocity and force data. The extracted friction-velocity curve is plotted in Fig. 11. According to these figures, the following observations may be made. • Apparently, the response lags the desired trajectory . This is primarily due to the closed-loop dynamics. The time lag may be reduced either by increasing the controller gain or by using a PD controller. Improved control of the system will be addressed in the next section. are some• The peaks of the actual position waveform what flat. This is in fact a backlash behavior in the closed-loop system which is caused by the deadband nonlinearity (mainly due to static friction) in the openloop system [4]. In other words, when the piston stops (as the direction of motion reverses), it arrives in the static friction regime. The servovalve current has to pass beyond some threshold value in order to overcome the static friction breakaway force. As a result, the piston will stay in the zero velocity condition for a finite duration of time. • The level of the dynamic frictional force is much less than what was observed in the quasistatic experiment. • A hysteresis behavior is present in the friction–velocity curve, which has been reported by other researchers as well [5]–[7]. • The chattering behavior of the estimated friction in the is due to the signum function neighborhoods of sgn in the Friedland–Park observer (see Fig. 4). Note that in the experiment outlined above, we have intentionally used a low feedback gain in order to observe these phenomena more clearly. To check (12) for the speed of convergence, a test was carried out on the recorded data. Suppose that at time s the observer is reset to zero. Numerical solution of (12) yields
(23)
(22) For the velocity profile defined by (20), this yields . With , the approximate pole location . Fig. 10 shows the result of applying would be
s the estimation error should reach 0.67% Thus, at of its initial value, which can be neglected. As shown in Fig. 12, computer simulation with the recorded data verifies this prediction.
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PARAMETERS
TABLE II TWO CONTROLLERS
OF THE
PID Control: Proportional gain K1 = 104 mA=m mA Derivative gain K2 = 110 m=s Acceleration Feedback Control: mA ; = 100; = 1 4 State feedback gains K = 1 m=s Observer gains kv = 500; ka = 1200 Threshold velocity Dv = 0:02m=s
0
proposed controller is obtained by adding an acceleration error term to a simple PD position controller, as shown in the block diagram of Fig. 14. The control law can be expressed as Fig. 12. server.
Investigation of the convergence rate of the Friedland–Park ob-
(24)
Fig. 13. On-line estimation of velocity and friction using the modified observer ( = 1:5; ka = 150; kv = 250).
C. On-Line Estimation of Friction and Velocity Fig. 13 shows the estimated velocity and friction, using the modified observer with and . These results are very close to the off-line estimation results given in Section V-B. VI. FRICTION-COMPENSATED POSITION TRACKING: METHODOLOGY In dc motors, the generated magnetic torque is directly proportional to the applied current [21]. Unlike dc motors, the is not proportional force generated by a hydraulic actuator to the current applied to its servovalve as significant actuator dynamics and nonlinearity would be present. Consequently, friction compensation in a hydraulic actuation system is not as straightforward as it is in a dc-motor-driven system. In the case of dc motors, one is able to conveniently add a correcting fraction of the estimated friction to the control effort so as to compensate for friction [5], [7], [8], [11]. The technique proposed herein employs the estimated frictional force to calculate acceleration. The acceleration signal is then used to further improve the tracking performance. A. Acceleration Feedback Control The estimated acceleration can be used in the feedback loop of the motion controller to decrease the tracking error. The
is the where is the current applied to the servovalve, desired value of the corresponding signal, and is a fixed set of the controller gains. Note that from velocity control viewpoint, the control law of (24) corresponds to a proportional-integral-derivative (PID) controller. It is in fact a state feedback controller where the acceleration term represents the internal state of the actuation system. Should there be no actuator dynamics, a simple PD position controller would form the state feedback; however, the actuator dynamics requires acceleration feedback, for satisfactory control. Acceleration feedback control has also been used by other researchers to obtain high performance hydraulic servomechanisms [19], [22], [23]. In [19], FitzSimons and Palazzolo used the root locus method to find the controller gains for a single-rod hydraulic actuator and presented simulation results. In [22], Welch discussed the quadratic resonance phenomenon that is generally observed in the transfer function relating the load velocity to the servovalve current. He then explained that acceleration feedback can be used to damp the hydraulic system so as to achieve a higher bandwidth. Welch assumed that the load is predominantly of inertial type (with no dissipation) and used pressure readings to calculate acceleration. By using linear analysis of the system dynamics, he verified the effectiveness of adding a pressure feedback term (as a minor feedback loop) to a conventional PD controller. He also suggested high-pass filtering of the pressure feedback signal, to allow full pressure feedback in the frequency range of the load resonance. Welch named his approach “derivative pressure feedback,” and presented experimental results using a hydraulic motor. A similar technique was employed by Matsuura et al. [23], for damping out the resonance of a hydraulic cylinder, in an industrial application. The approach presented by us can also be considered a pressure feedback technique (see Fig. 14), with the following unique features. • Frictional effects are taken into account. • Controller gains are chosen using a well-established heuristic method. In [13], Studenny et al. proposed an approximate method to choose the gains of an acceleration feedback controller. is Accordingly, in a stable closed-loop system, as the gain made large, the dynamics of the position error
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Fig. 14.
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Block diagram of the acceleration feedback control system.
C. Selection of Observer Gains As a rule of thumb, both the velocity and friction observers should be made at least four times faster than the closedloop system. For the feedback gains chosen as in (27), the approximate closed-loop poles are located at (30)
Fig. 15.
Then, observer poles chosen gain is (22), we choose
Ideal response to the desired position trajectory.
. For the velocity observer, the and for the friction observer, using (31)
may be approximately represented by (25) which corresponds to the following second-order characteristic equation for the closed-loop system: with
(26)
B. Selection of Controller Gains The heuristic method of Studenny et al. [13] is used here to assign the gains and . Controller tuning might be required then, to achieve further improvement in the response. By choosing the desired natural frequency and damping ratio as rad/s and , (26) gives (27) The gain can be chosen using the nominal current of the servovalve ( mA). The choice of mA/mm
mA/m
(28)
yields mA m/s
(29)
Given a typical desired velocity profile, from (31).
can be computed
D. Quintic Polynomial Approach According to [24], a desired trajectory with smooth position, velocity, acceleration, and jerk components can be generated using a quintic (fifth-order) polynomial. With this choice, the motion noise of the actuator is also reduced. Details on choosing the coefficients of the polynomials to meet system specifications and limitations are given in [24]. In the present application, it is required to move the object (cutter blade assembly) from its initial stationary position to the desired position which is also stationary. The transient trajectory is chosen as a quintic polynomial. E. Ideal Response Ideally, the closed-loop system should behave as a secondorder linear system with the following transfer function: (32) Fig. 15 shows the simulated response of the transfer function of (32) to the desired quintic position trajectory that will be used in the experiments. The simulation result shows that the and are suitable. According to Fig. 15, chosen values for
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Fig. 16.
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(a)
(a)
(b)
(b)
(c)
(c)
Position tracking without friction compensation.
Fig. 17. Position tracking with friction compensation.
in the ideal case, the closed-loop system exhibits a time lag of 14 ms. This value was found by calculating the time shift of the response which would result in the minimum root-meansquare (rms) value of the tracking error (0.047 mm). This rms position tracking error is defined as follows: (33) where is the response lag. Note that the response lag is not important in our application, provided that the final time is within specifications. The most significant issue is the closeness of the shape of the actual trajectory to the desired trajectory.
where mA is the threshold value of the input current. This term is included to compensate for the deadband nonlinearity of the open-loop system that was discussed in Section V-B. Figs. 16 and 17 show the experimental results. Fig. 16 corresponds to the control law without friction compensation (a well-tuned PD controller); as given by (35) and Fig. 17 corresponds to the control law with friction compensation (acceleration feedback controller); as given by (36)
VII. FRICTION-COMPENSATED POSITION TRACKING: EXPERIMENTAL RESULTS Tracking performance of the electrohydraulic actuator with and without friction compensation is investigated now, through experimentation. The digital controllers are implemented at a sampling frequency of kHz. To reduce the steady-state position tracking error, the following correction term is also added to the control effort of both controllers: (34)
For the desired range of motions, typically, we have . Consequently, using (31), a value of is chosen for the friction observer gain. The threshold velocity is chosen m/s, using a trial and error approach. equal to The parameters of both controllers are listed in Table II. Note that the tuned gain of the acceleration feedback controller is different from its initial value given by (27). Figs. 16(a) and 17(a) show that there is a slight lag (12 ms) in the responses, which is primarily due to the closed-loop
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Comparison of the position tracking error profiles.
Fig. 20. Comparison of the position tracking performance for a desired trajectory that approaches the motion limits.
Finally, in order to investigate the control performance when the actuation system exhibits increased nonlinear behavior, we present the experimental results with a trajectory that passes closer to the ends of the allowable motion domain. Fig. 20 compares the position responses using the two controllers. With the new desired trajectory, the piston reaches within 1.27 mm of the limits of the motion. A longer transition time has been used in this experiment, to prevent exceeding the maximum achievable speed in the desired trajectory; otherwise, saturation would occur. Once again, it is observed that the friction-compensating controller has a superior performance. VIII. CONCLUSIONS
Fig. 19.
Comparison of the velocity tracking error profiles.
dynamics of the system. As discussed earlier, ideally, the system response should have a time lag of 14 ms. Also, note that there are more high-frequency components in Fig. 17(c) than in Fig. 16(c), which is a result of including the acceleration error term in the control law. By shifting the position signals through 12 ms to compensate for the time lag, one can compare the tracking errors quantitatively. Fig. 18 shows the position tracking error signals. According to the rms values shown in this figure, using the friction-compensating controller results in reduction of the position error by a factor of 1.69. Likewise, Fig. 19 shows the velocity tracking error signals. The rms values show that using the friction-compensating controller results in reduction of the velocity error by a factor of 2.36.
Fast and accurate position control of a typical electrohydraulic manipulator, under friction nonlinearities, was considered in this paper. The manipulator considered is an integral part of an automated fish processing machine. In this machine, it is required to quickly move the object (cutter blade assembly) to its new position, as smoothly and as precisely as possible. Experiments showed that considerable static and dynamic friction exists in the prototype system. The nonlinear observer of Friedland and Park was used to estimate the dynamic friction. Convergence speed of the observer was studied, which was found to be useful for choosing the observer parameters. The Friedland–Park observer was combined with a first-order velocity filter, for real-time estimation of friction. A method was presented to convert the friction estimator to an acceleration observer. A novel control strategy for compensation of friction in electrohydraulic actuators was proposed that combines friction estimation with an acceleration feedback control law. The experimental results showed that, for a typical trajectory, the RMS position and velocity tracking errors were reduced by factors of 1.69 and 2.36, respectively. The approach pre-
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sented in this paper is quite general and may be applied to any servomechanism with significant actuator dynamics and friction.
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[23] T. Matsuura, N. Shigematsu, K. Nakashima, and K. Moribe, “Hydraulic screwdown control system for mandrel mill,” IEEE Trans. Ind. Applicat., vol. 30, pp. 568–572, May/June 1994. [24] R. L. Andersson, A Robot Ping-Pong Player: Experiments in Real-Time Intelligent Control. Cambridge, MA: MIT Press, 1988, pp. 94–95.
ACKNOWLEDGMENT The authors wish to thank colleague Dr. E. A. Croft for her careful reading of the manuscript. REFERENCES [1] C. W. de Silva, “Research laboratory for fish processing automation,” Robot. Comput.-Integrated Manufacturing, vol. 9, no. 1, pp. 49–60, 1992. [2] S. Tafazoli, C. W. de Silva, and P. D. Lawrence, “Friction estimation in a planar electrohydraulic manipulator,” in Proc. Amer. Contr. Conf., Seattle, WA, June 1995, vol. 5, pp. 3294–3298. , “Position and force control of an electrohydraulic manipulator [3] in the presence of friction,” in Proc. IEEE Conf. Syst., Man, Cybern., Vancouver, BC, Oct. 1995, vol. 2, pp. 1687–1692. [4] H. E. Merritt, Hydraulic Control Systems. New York: Wiley, 1967. [5] B. Armstrong-H´elouvry, P. Dupont, and C. Canudas de Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol. 30, no. 7, pp. 1083–1138, May 1994. ˚ om, and P. Lischinsky, “A [6] C. Canudas de Wit, H. Olsson, K. J. Astr¨ new model for control of systems with friction,” IEEE Trans. Automat. Contr., vol. 40, pp. 419–425, Mar. 1995. [7] B. Armstrong-H´elouvry, Control of Machines with Friction. Boston, MA: Kluwer, 1991. [8] B. Friedland and Y. J. Park, “On adaptive friction compensation,” IEEE Trans. Automat. Contr., vol. 37, pp. 1609–1612, Oct. 1992. [9] B. Friedland and S. Mentzelopoulou, “On adaptive friction compensation without velocity measurement,” in Proc. IEEE Conf. Contr. Applicat., Dayton, OH, Sept. 1992, vol. 2, pp. 1076–1081. [10] S. Mentzelopoulou and B. Friedland, “Experimental evaluation of friction estimation and compensation techniques,” in Proc. Amer. Contr. Conf., Baltimore, MD, June 1994, vol. 3, pp. 3132–3136. [11] J. Amin, B. Friedland, and A. Harnoy, “Implementation of a friction estimation and compensation technique,” IEEE Contr. Syst. Mag., vol. 17, no. 4, pp. 71–76, Aug. 1997. [12] B. de Jager, “Acceleration assisted tracking control,” IEEE Contr. Syst. Mag., vol. 14, no. 5, pp. 20–27, Oct. 1994. [13] J. Studenny, P. R. B´elanger, and L. K. Daneshmand, “A digital implementation of the acceleration feedback control law on a PUMA 560 manipulator,” in Proc. 30th Conf. Decision Contr., Brighton, U.K., Dec. 1991, vol. 3, pp. 2639–2648. [14] D. Vischer and O. Khatib, “Design and development of highperformance torque-controlled joints,” IEEE Trans. Robot. Automat., vol. 11, no. 4, pp. 537–544, Aug. 1995. [15] L. E. Pfeffer, O. Khatib, and J. Hake, “Joint torque sensory feedback in the control of a PUMA manipulator,” IEEE Trans. Robot. Automat., vol. 5, pp. 418–425, Aug. 1989. [16] J. Y. S. Luh, W. D. Fisher, and R. P. C. Paul, “Joint torque control by a direct feedback for industrial robots,” IEEE Trans. Automat. Contr., vol. AC-28, pp. 153–161, Feb. 1983. [17] G. Vossoughi and M. Donath, “Dynamic feedback linearization for electrohydraulically actuated control systems,” J. Dynamic Syst., Measurement, Contr., Trans. ASME, vol. 117, pp. 468–477, Dec. 1995. [18] P. M. FitzSimons and J. J. Palazolo, “PART I: Modeling of a one-degreeof-freedom active hydraulic mount,” J. Dynamic Syst., Measurement, Contr., Trans. ASME, vol. 118, no. 3, pp. 439–442, Sept. 1996. [19] , “PART II: Control of a one-degree-of-freedom active hydraulic mount,” J. Dynamic Syst., Measurement, Contr., Trans. ASME, vol. 118, no. 3, pp. 443–448, Sept. 1996. [20] D. Karnopp, “Computer simulation of stick-slip friction in mechanical dynamic systems,” J. Dynamic Syst., Measurement, Contr., Trans. ASME, vol. 107, pp. 100–103, Mar. 1985. [21] C. W. de Silva, Control Sensors and Actuators. Englewood Cliffs, NJ: Prentice-Hall, 1989. [22] T. R. Welch, “The use of derivative pressure feedback in high performance hydraulic servomechanisms,” J. Eng. Ind., Trans. ASME, vol. 84, pp. 8–14, Feb. 1962.
Shahram Tafazoli (S’95–M’98) received the B.A.Sc. and M.A.Sc. (with honors) degrees in electrical engineering from Sharif University of Technology, Tehran, Iran, 1989 and 1991, respectively. He received the Ph.D. degree, also in electrical engineering, from the University of British Columbia (UBC), Vancouver, Canada, in January 1997. Currently, he is collaborating with the Robotics and Control Laboratory of UBC as a Research and Development Consultant. His research interests include friction estimation and compensation in mechanical systems, position and force control of robotic manipulators, and teleoperation. He has an application emphasis on teleoperated control and real-time payload estimation of the heavy-duty mobile hydraulic manipulators, such as excavators. Dr. Tafazoli held various fellowships and awards during his education and he was the Vice Chairman of the IEEE Control Systems Society, Vancouver Chapter, in 1995. He is also a member of the ASME.
Clarence W. de Silva (S’75–M’78–SM’85–F’98) received the Ph.D. degree in dynamic systems and control from the Massachusetts Institute of Technology, Cambridge, in 1978. Since 1988, he has been Professor of Mechanical Engineering and holder of the NSERC-BC Packers Chair in Industrial Automation at the University of British Columbia, Vancouver. He has authored 13 books and more than 100 journal papers, the most recent being Intelligent Control: Fuzzy Logic Applications (Boca Raton, FL: CRC, 1995). Dr. de Silva has been a Lilly Fellow, NASA/ASEE Fellow, Fullbright Fellow, ASI Fellow, Killam Fellow, and an ASME Fellow. He has served on editorial boards of 12 international journals including IEEE and ASME TRANSACTIONS, and is the Regional Editor, North America for the IFACaffiliated International Journal of Intelligent Real-Time Automation and Editor-in-Cheif of the International Journal of Knowledge-Based Intelligent Engineering Systems. He has received the Outstanding Contribution Award from the IEEE SMC Society, and the Meritorious Achievement Award from the Association of Professional Engineers of British Columbia.
Peter D. Lawrence (S’64–M’73) received the B.A.Sc. degree in electrical engineering from the University of Toronto, Canada, in 1965, the Masters degree in biomedical engineering from the University of Saskatchewan, Canada, in 1967, and the Ph.D. degree in computing and information science from Case Western Reserve University, Cleveland, OH, in 1970. He worked as a Guest Researcher at Chalmers University’s Applied Electronics Department in Goteborg, Sweden, between 1970 and 1972, and between 1972 and 1974 as a Research Staff Member and Lecturer in the Mechanical Engineering Department of the Massachusetts Institute of Technology, Cambridge. Since 1974, he has been with the University of British Columbia and is a Professor in the Department of Electrical and Computer Engineering. His main research interests include the application of real-time computing in the control interface between humans and machines, image processing, and mobile hydraulic machine modeling and control. Dr. Lawrence is a member of SAE and a registered Professional Engineer.
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Tracking Control of an Electrohydraulic Manipulator in the Presence of Friction Shahram Tafazoli, Member, IEEE, Clarence W. de Silva, Fellow, IEEE, and Peter D. Lawrence, Member, IEEE
Abstract—Analysis and estimation of friction and compensation for its effects in the control of an electrohydraulic manipulator is addressed in this paper. The specific hydraulic manipulator is an integral part of an automated fish processing machine which has been developed in our laboratory. The analysis reveals that considerable static and dynamic friction exists in the system. An available nonlinear observer for Coulomb friction, is modified to simultaneously estimate friction, velocity, and acceleration. A novel observer-based friction compensating control strategy is developed for improved tracking performance of the manipulator. The approach is based on acceleration feedback control. Experimental investigations show that this controller significantly outperforms a conventional, proportional-plus-derivative (PD) controller. The general approach presented in this paper, may be applied to compensate for friction in any servomechanism, particularly when the actuator dynamics is not negligible. Index Terms—Acceleration control, error analysis, mechanical factors, observers, parameter estimation, servosystems, tracking.
I. INTRODUCTION
A
N industrial automation laboratory has been established primarily for research and development of advanced technology in automation of the fish processing industry [1]. In this laboratory, an automated machine for mechanical processing of salmon has been developed. In this industrial prototype, the fish are transported using a conveyor system. When a fish reaches the cutter blade, its image is taken by a digital camera. A computer analyzes the image and ) for removing generates the desired position coordinates ( the head, with minimal wastage of meat. The control system of two independent ( and ) electrohydraulic actuators then performs precise positioning of the cutter blade assembly. In order to meet the process requirements, this motion has to be fast, accurate, smooth, and nonoscillatory. After the blade is moved to the desired position in the horizontal plane, a pneumatic actuator that moves the blade in the vertical ( ) direction is activated, in order to remove the fish head. Note that the cutter positioning mechanism is in fact a Cartesian
Manuscript received November 3, 1995. Recommended by Associate Editor, F. Svaricek. C. W. de Silva was supported by the NSERC Industrial Automation Chair Grant 5-80595 and B.C. Advanced Systems Institute Fellowship Grant 5-56296. P. D. Lawrence was supported by NSERC NCE IRIS Grant 5-58305. Additional funding was provided by B.C. Packers, Ltd. and Garfield Weston Foundation. S. Tafazoli and P. D. Lawrence are with the Electrical and Computer Engineering Department, University of British Columbia, Vancouver, BC, Canada V6T 1Z4. C. W. de Silva is with the Mechanical Engineering Department, University of British Columbia, Vancouver, BC, Canada V6T 1Z4. Publisher Item Identifier S 1063-6536(98)03090-5.
electrohydraulic manipulator, and is an integral part of the fish processing machine. The sliding surfaces in the two axes generate frictional forces. Since friction can degrade the system performance, it is undertaken to identify it, with the objective of compensation for its effects in closed-loop control. Analysis and on-line estimation of the frictional force in the system has been studied in our previous work [2]. Therein, we found that a considerable amount of static and dynamic friction exists in the prototype system. The frictional force arises primarily due to the greaselubricated metal guideways on which the blade assembly slides, and also due to rubbing of the hydraulic pistons against their cylinder walls. In [2], a nonlinear observer was developed that simultaneously estimates the dynamic friction and velocity. Position and force control of the electrohydraulic actuator was studied subsequently, in [3]. In this paper, the previous investigations are combined and enhanced in order to gain a thorough insight into both the behavior of friction in the prototype system and the performance of the friction-compensating controller. Furthermore, a practical method is developed for choosing the gains of the controller and the associated observer. Even though a specific machine (an electrohydraulic servomechanism) is studied here, the overall control strategy that is developed is quite general and can be applied to any mechanical actuation system, especially when the actuator dynamics is not negligible. Friction is a complex, natural phenomenon which is present in virtually all mechanical control systems including electric, pneumatic, and hydraulic actuators. It can pose a serious challenge to achieving high-quality performance of these machines. In particular, it plays an important albeit damaging role in hydraulic control systems [4]. Friction can lead to tracking errors, limit cycle oscillations, and undesirable stick-slip motion. As a result, modeling of friction and compensation for its effects has received considerable attention from researchers [5]–[8]. In the present paper, we will particularly focus on the observer-based friction estimation and compensation technique that has been proposed by Friedland et al. [8]–[11]. Acceleration-assisted tracking control has been discussed by de Jager [12], where a performance improvement by a factor of up to 1.5 is reported in the particular application. According to de Jager, acceleration measurement in mechanical control systems can be used to decrease the tracking error, improve the robustness for modeling errors, and reduce the effective friction. Studenny et al. [13] have successfully applied acceleration feedback control to robotic manipulators. An alternative method for friction compensation is to measure the transmitted
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TABLE I PHYSICAL PARAMETERS OF THE PROTOTYPE MANIPULATOR
Fig. 1. The high-speed electrohydraulic positioning system (for one axis).
force (torque) and close a force (torque) feedback loop. The approach, termed joint torque sensory feedback, has been applied for force control of robotic manipulators [14]–[16]. In the system that is considered in the present paper, there is no sensor to directly measure the acceleration or the transmitted force. Instead of direct measurement, acceleration will be computed from the estimated friction. Then, by combining friction estimation and acceleration feedback control, an adaptive friction compensation technique is established. This paper is organized into several sections in the following manner: In Section II, the experimental system is briefly described. In Section III, a general state-space model is derived for the electrohydraulic manipulator, primarily to show the nonlinear nature of the actuator dynamics. In Section IV, the nonlinear observer for friction parameter, proposed by Friedland and Park [8], is discussed and its convergence rate is analyzed. In this section, a new combined observer is developed for simultaneous estimation of friction, velocity, and acceleration. In Section V, the experimental results on friction analysis and estimation are presented. In Section VI, a new friction-compensating control strategy is introduced which is based on acceleration feedback control. In Section VII, performance of the proposed controller is experimentally investigated with comparison to that of a conventional proportionalplus-derivative (PD) controller. Conclusions are outlined in Section VIII. II. THE EXPERIMENTAL SYSTEM The prototype electrohydraulic manipulator is of Cartesian type with two orthogonal prismatic joints. It has two control inputs which are the currents applied to the electrohydraulic proportional valves for the and directions. Since the two degrees of freedom in the manipulator are quite similar and also independent with regard to operation, only the direction is studied in this paper. Fig. 1 shows the simplified schematic diagram of the manipulator, for the axis. The servovalve consists of a pilot stage with torque-motoractuated flapper and a spring-centered boost stage with double sliding spool arrangement. The differential pressure from the
pilot is applied across both spools of the boost stage and is balanced by the centering springs, producing a spool position that is proportional to the differential pressure and therefore to the input current. The nominal full flow current of this mA. two-stage servovalve is A Temposonics linear displacement transducer is installed at the head of the hydraulic cylinder. This industrial magnetostrictive sensor precisely senses the position of an external magnet, which is connected to the piston, from the time interval between an interrogation pulse and a return pulse. The specific sensor that is used has a resolution of 0.025 mm and its measurement update time is less than 1 ms. Two gage-pressure transducers are also installed on the head and rod sides of and the cylinder, in order to measure the fluid pressures , respectively. Note that there is no velocity or acceleration sensor in the system. All the sensor outputs (including pressure transducers) are low-pass filtered by first-order antialiasing kHz is used RC filters. A sampling frequency of to implement the digital controllers; accordingly, the cutoff frequency of the filters is set at 200 Hz. A 12–bit PC-based data acquisition system is used for data collection and graphical display. The programs have been written in C++. The physical parameter values of the prototype manipulator are listed in Table I. III. MODELING OF THE ELECTROHYDRAULIC SYSTEM A dynamic model is derived in this section for the electrohydraulic manipulator. This analysis is intended primarily to emphasize the nonlinear nature of the actuator dynamics. Similar approaches to modeling of asymmetric hydraulic actuators have been reported in [17] and [18]. Primarily, there are three types of nonlinearities in the servovalve system: namely, the basic flow equation through an orifice, the hysteresis resulting from electromagnetic characteristics of the torque motor, and the flow forces on the valve spools [4], [17]. Displacement of the valve spools from their null position creates a pressure difference across the single-rod hydraulic actuator and the resulting fluid flow causes a change in the position of the piston. The force applied by the actuator can be calculated from cylinder pressures. Newton’s second law applied to the actuator yields (1) where force,
is the actuator force, is the opposing frictional is the piston displacement, and is the equivalent
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Fig. 2. A simplified block diagram of the system.
moving mass in the direction. The amount of fluid flow to and from the rod-side of the cylinder the head-side is a function of both the valve spool position and cylinder pressures. Assuming that the valve is of critical-center type with matched and symmetrical ports (see [4] for definitions), the relationship can be expressed in the following form [18]:
Fig. 3. System model with Coulomb friction.
required in the control strategy that is proposed in the present paper. IV. ESTIMATION OF FRICTION, VELOCITY, AND ACCELERATION
(2) A. The Friedland–Park Friction Observer [8] is the valve spool displacement, is a fixed gain, where is the supply pressure, and is a switching function defined as below (3) . Furthermore, using the continuity principle, and taking fluid compressibility into account, one can write [4] (4) is where is the effective bulk modulus of the fluid and the volume of the fluid inside each of the hoses that connect the servovalve to the actuator. Next, define a state vector for the system as follows: (5)
In a dynamic experiment, an observer is needed to estimate the frictional force. Since the force applied by the actuator can be easily calculated from pressure readings, the nonlinear observer proposed by Friedland and Park in [8] for Coulomb friction, appears to be suitable in the present application. The Friedland–Park observer is briefly discussed in the following. Note that their approach is reformulated here in order to include the object mass . System dynamics is assumed to be as shown by the block diagram in Fig. 3. In this figure, the block denoted by “NL” represents the nonlinear friction element. The frictional force is assumed to be of classical Coulomb type, as given by (7) To estimate the Coulomb friction parameter , Friedland and Park proposed the following nonlinear observer:
By combining (1), (2), and (4), one arrives at the following nonlinear state-space model of the system:
(8) is the estimated Coulomb friction parameter, where denotes the state variable of the observer, and the design and the exponent . The parameters are the gain variable represents the estimated acceleration and is given by
(6) (9) where
is the estimated Coulomb friction; i.e., sgn
A simplified block diagram of the hydraulic manipulator is shown in Fig. 2. In this diagram, the “actuator dynamics” block represents the nonlinear hydraulic actuation system. Servovalve dynamics can also be included in this block. According to the state-space equations, the actuator dynamics depends on the position and velocity of the piston. Vossoughi and Donath [17] have discussed various nonlinearities that are present in electrohydraulic systems. They used feedback linearization to compensate for these effects in the control design. Knowledge of the manipulator dynamics is not
(10)
Fig. 4 shows a block diagram of the Friedland–Park nonlinear observer. Defining estimation error as , where is the true friction parameter, Friedland and Park have shown that the error dynamics is given by (11) Therefore, for a fixed true parameter ( ), the estimation error converges asymptotically to zero if and is bounded away from zero. By simulation, Friedland and Park
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Fig. 5. The velocity filter.
Fig. 4. The Friedland–Park Coulomb friction observer.
have also confirmed that the observer works well for more is a nonlinear function complicated friction models where of velocity. B. Convergence Rate of the Friedland–Park Observer Convergence rate of the Coulomb friction observer that has been proposed by Friedland and Park, is analyzed now. This is an original contribution of the present paper. According to (11), the estimation error has linear, time-varying dynamics. ), explicit solution of Assuming a fixed true parameter ( , gives (11) for an initial estimation error (12)
As discussed in [2], the observer proposed by Friedland and Menzelopoulou must be modified in order to accommodate the static friction component in our system. In the modified observer, velocity estimation is performed independently of the friction estimation. The velocity observer is in fact a firstorder low-pass filtered differentiator, and is represented by the is the observer transfer function given in Fig. 5, where gain. The estimated velocity is then used in the Friedland–Park algorithm. Implementation of the modified observer for friction estimation and compensation in a dc-motor-driven apparatus has been recently reported in [11], where the authors use the name “Tafazoli modification” for the proposed observer. Their experimental investigations show that the performance of the observer is very promising, such that friction estimates obtained compare favorably to the actually measured friction, and significant tracking improvement is obtained by compensating for the estimated friction in the control law. Since the estimation update is poor at low velocities, the observer estimates dynamic friction only. Nevertheless, static friction has to be taken into account for estimating the net friction. The following interpolating function will be used for this purpose: (15)
Thus, nonzero velocity drives the estimation error to zero. Equation (12) can be written in the following form, to establish an analogy with first-order linear time-invariant observers
with (13) It follows that, the approximate pole of the observer is located at (14) denotes the mean value of the corresponding variwhere able. Equation (14) can be used to choose the gain and the exponent for a desired rate of convergence. C. Simultaneous Estimation of Friction, Velocity, and Acceleration In [9], Friedland and Mentzelopoulou developed a combined observer for friction and velocity in the absence of direct velocity measurement in the system. The authors employed an observer for velocity estimation and then used this estimate in the Friedland–Park algorithm. They showed local asymptotic stability for both velocity and friction estimations.
This empirical function is adopted from [5] and [6], where it has been used in a different context. In (15), is the applied force, is the net friction, is the Coulomb (dynamic) friction, and is the threshold velocity. A small neighborhood of zero velocity is defined by , as in the model proposed by Karnopp [20]. According to (15), outside this neighborhood, friction is mainly dynamic in nature (a function of velocity). Inside the neighborhood, velocity is considered negligible, and static friction is dominant (a function of the applied force). This formulation is used to account for noise and numerical inaccuracy of the estimated velocity. Thus, the net frictional force can be computed from the following equation: (16) Likewise, the estimated acceleration would be (17) The variable in (9) represents the acceleration signal provided that the velocity is away from zero. By combining (9) and (17), we obtain (18) Thus, the estimated acceleration is an enhanced version of , as static friction is taken into account in its calculation. Fig. 6
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Fig. 6. The combined observer. Fig. 7. The quasistatic sinusoidal response of the open-loop system.
shows the final form of our new observer for velocity, friction, and acceleration. The design parameters for the observer are , , , and . V. ANALYSIS AND ESTIMATION OF FRICTION: EXPERIMENTAL RESULTS In this section, the characteristics of friction at very low velocities is investigated. Also, off-line and on-line estimation of friction and velocity in the prototype manipulator, are addressed. The Euler approximation is used for digital implementation of the nonlinear observers.
Fig. 8. Friction-velocity characteristic curve at low velocities.
A. Friction-Velocity Characteristic at Very Low Velocities In a quasistatic experiment, when the piston moves very slowly, the inertial force in (1) can be neglected compared to . Then, the applied force is a good representation of the frictional force. A sinusoidal current with an amplitude of 2.5 mA (7% of the rated current) and a period of 100 s was applied to the servovalve. The pressures and the piston position were sampled every 2 s. Note that in this system, piston movement is quite limited. In view of this, the applied current to the servovalve, in open-loop experiments, should be small. To estimate velocity from position signal, a digital filter was employed off-line, which is in fact a high-order, delay-free, low-pass-filtered differentiator [2]. Fig. 7 shows the result of this test. According to this figure, there is a significant amount of static friction that causes deadband nonlinearity in the open-loop system [4]. The effect of this nonlinearity on the closed-loop response will be explained later. The measured force and the estimated velocity were used to obtain the steady-state, friction-velocity characteristic, and is plotted in Fig. 8. Note that only one response cycle is used to obtain this curve. The curve was found to be slightly
different for different cycles, which is due to the random characteristics of friction. It is evident from Fig. 8 that the friction behavior resembles that of classical Coulomb friction with some hysteresis effect. However, the transition from static friction to a dynamic friction regime is not abrupt, contrary to the classical friction model [5]–[7]. Finally, acceleration was estimated by applying a similar digital filter to the velocity signal (to obtain ). As shown in is quite negligible compared to Fig. 9, the inertial force the applied force . B. Off-Line Estimation of Friction and Velocity In a dynamic experiment, the Friedland–Park nonlinear observer can be used off-line, employing the digital differentiator discussed in Section V-A to estimate velocity. A digital proportional controller (with a gain of mA/m) was used to close the feedback loop, at a sampling frequency of kHz. The desired position trajectory was chosen as m
(19)
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Fig. 9. The estimated inertial force in the quasistatic experiment.
Fig. 11. Estimated dynamic friction as a function of velocity.
Fig. 10. Off-line estimation of Coulomb friction using the Friedland–Park observer ( = 1:5; ka = 150).
which gives the following expression for the desired velocity: m/s
(20)
To study the bandwidth of the observer, assuming that the tracking error is not very large, one can replace the actual in (14) with the desired velocity velocity (21) , in order to obtain a high speed of convergence, Since . it is desirable to choose the exponent in the range , the expression for the observer pole becomes With
the Friedland–Park observer (with ) to the recorded velocity and force data. The extracted friction-velocity curve is plotted in Fig. 11. According to these figures, the following observations may be made. • Apparently, the response lags the desired trajectory . This is primarily due to the closed-loop dynamics. The time lag may be reduced either by increasing the controller gain or by using a PD controller. Improved control of the system will be addressed in the next section. are some• The peaks of the actual position waveform what flat. This is in fact a backlash behavior in the closed-loop system which is caused by the deadband nonlinearity (mainly due to static friction) in the openloop system [4]. In other words, when the piston stops (as the direction of motion reverses), it arrives in the static friction regime. The servovalve current has to pass beyond some threshold value in order to overcome the static friction breakaway force. As a result, the piston will stay in the zero velocity condition for a finite duration of time. • The level of the dynamic frictional force is much less than what was observed in the quasistatic experiment. • A hysteresis behavior is present in the friction–velocity curve, which has been reported by other researchers as well [5]–[7]. • The chattering behavior of the estimated friction in the is due to the signum function neighborhoods of sgn in the Friedland–Park observer (see Fig. 4). Note that in the experiment outlined above, we have intentionally used a low feedback gain in order to observe these phenomena more clearly. To check (12) for the speed of convergence, a test was carried out on the recorded data. Suppose that at time s the observer is reset to zero. Numerical solution of (12) yields
(23)
(22) For the velocity profile defined by (20), this yields . With , the approximate pole location . Fig. 10 shows the result of applying would be
s the estimation error should reach 0.67% Thus, at of its initial value, which can be neglected. As shown in Fig. 12, computer simulation with the recorded data verifies this prediction.
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PARAMETERS
TABLE II TWO CONTROLLERS
OF THE
PID Control: Proportional gain K1 = 104 mA=m mA Derivative gain K2 = 110 m=s Acceleration Feedback Control: mA ; = 100; = 1 4 State feedback gains K = 1 m=s Observer gains kv = 500; ka = 1200 Threshold velocity Dv = 0:02m=s
0
proposed controller is obtained by adding an acceleration error term to a simple PD position controller, as shown in the block diagram of Fig. 14. The control law can be expressed as Fig. 12. server.
Investigation of the convergence rate of the Friedland–Park ob-
(24)
Fig. 13. On-line estimation of velocity and friction using the modified observer ( = 1:5; ka = 150; kv = 250).
C. On-Line Estimation of Friction and Velocity Fig. 13 shows the estimated velocity and friction, using the modified observer with and . These results are very close to the off-line estimation results given in Section V-B. VI. FRICTION-COMPENSATED POSITION TRACKING: METHODOLOGY In dc motors, the generated magnetic torque is directly proportional to the applied current [21]. Unlike dc motors, the is not proportional force generated by a hydraulic actuator to the current applied to its servovalve as significant actuator dynamics and nonlinearity would be present. Consequently, friction compensation in a hydraulic actuation system is not as straightforward as it is in a dc-motor-driven system. In the case of dc motors, one is able to conveniently add a correcting fraction of the estimated friction to the control effort so as to compensate for friction [5], [7], [8], [11]. The technique proposed herein employs the estimated frictional force to calculate acceleration. The acceleration signal is then used to further improve the tracking performance. A. Acceleration Feedback Control The estimated acceleration can be used in the feedback loop of the motion controller to decrease the tracking error. The
is the where is the current applied to the servovalve, desired value of the corresponding signal, and is a fixed set of the controller gains. Note that from velocity control viewpoint, the control law of (24) corresponds to a proportional-integral-derivative (PID) controller. It is in fact a state feedback controller where the acceleration term represents the internal state of the actuation system. Should there be no actuator dynamics, a simple PD position controller would form the state feedback; however, the actuator dynamics requires acceleration feedback, for satisfactory control. Acceleration feedback control has also been used by other researchers to obtain high performance hydraulic servomechanisms [19], [22], [23]. In [19], FitzSimons and Palazzolo used the root locus method to find the controller gains for a single-rod hydraulic actuator and presented simulation results. In [22], Welch discussed the quadratic resonance phenomenon that is generally observed in the transfer function relating the load velocity to the servovalve current. He then explained that acceleration feedback can be used to damp the hydraulic system so as to achieve a higher bandwidth. Welch assumed that the load is predominantly of inertial type (with no dissipation) and used pressure readings to calculate acceleration. By using linear analysis of the system dynamics, he verified the effectiveness of adding a pressure feedback term (as a minor feedback loop) to a conventional PD controller. He also suggested high-pass filtering of the pressure feedback signal, to allow full pressure feedback in the frequency range of the load resonance. Welch named his approach “derivative pressure feedback,” and presented experimental results using a hydraulic motor. A similar technique was employed by Matsuura et al. [23], for damping out the resonance of a hydraulic cylinder, in an industrial application. The approach presented by us can also be considered a pressure feedback technique (see Fig. 14), with the following unique features. • Frictional effects are taken into account. • Controller gains are chosen using a well-established heuristic method. In [13], Studenny et al. proposed an approximate method to choose the gains of an acceleration feedback controller. is Accordingly, in a stable closed-loop system, as the gain made large, the dynamics of the position error
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Block diagram of the acceleration feedback control system.
C. Selection of Observer Gains As a rule of thumb, both the velocity and friction observers should be made at least four times faster than the closedloop system. For the feedback gains chosen as in (27), the approximate closed-loop poles are located at (30)
Fig. 15.
Then, observer poles chosen gain is (22), we choose
Ideal response to the desired position trajectory.
. For the velocity observer, the and for the friction observer, using (31)
may be approximately represented by (25) which corresponds to the following second-order characteristic equation for the closed-loop system: with
(26)
B. Selection of Controller Gains The heuristic method of Studenny et al. [13] is used here to assign the gains and . Controller tuning might be required then, to achieve further improvement in the response. By choosing the desired natural frequency and damping ratio as rad/s and , (26) gives (27) The gain can be chosen using the nominal current of the servovalve ( mA). The choice of mA/mm
mA/m
(28)
yields mA m/s
(29)
Given a typical desired velocity profile, from (31).
can be computed
D. Quintic Polynomial Approach According to [24], a desired trajectory with smooth position, velocity, acceleration, and jerk components can be generated using a quintic (fifth-order) polynomial. With this choice, the motion noise of the actuator is also reduced. Details on choosing the coefficients of the polynomials to meet system specifications and limitations are given in [24]. In the present application, it is required to move the object (cutter blade assembly) from its initial stationary position to the desired position which is also stationary. The transient trajectory is chosen as a quintic polynomial. E. Ideal Response Ideally, the closed-loop system should behave as a secondorder linear system with the following transfer function: (32) Fig. 15 shows the simulated response of the transfer function of (32) to the desired quintic position trajectory that will be used in the experiments. The simulation result shows that the and are suitable. According to Fig. 15, chosen values for
TAFAZOLI et al.: TRACKING CONTROL OF AN ELECTROHYDRAULIC MANIPULATOR
Fig. 16.
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(a)
(a)
(b)
(b)
(c)
(c)
Position tracking without friction compensation.
Fig. 17. Position tracking with friction compensation.
in the ideal case, the closed-loop system exhibits a time lag of 14 ms. This value was found by calculating the time shift of the response which would result in the minimum root-meansquare (rms) value of the tracking error (0.047 mm). This rms position tracking error is defined as follows: (33) where is the response lag. Note that the response lag is not important in our application, provided that the final time is within specifications. The most significant issue is the closeness of the shape of the actual trajectory to the desired trajectory.
where mA is the threshold value of the input current. This term is included to compensate for the deadband nonlinearity of the open-loop system that was discussed in Section V-B. Figs. 16 and 17 show the experimental results. Fig. 16 corresponds to the control law without friction compensation (a well-tuned PD controller); as given by (35) and Fig. 17 corresponds to the control law with friction compensation (acceleration feedback controller); as given by (36)
VII. FRICTION-COMPENSATED POSITION TRACKING: EXPERIMENTAL RESULTS Tracking performance of the electrohydraulic actuator with and without friction compensation is investigated now, through experimentation. The digital controllers are implemented at a sampling frequency of kHz. To reduce the steady-state position tracking error, the following correction term is also added to the control effort of both controllers: (34)
For the desired range of motions, typically, we have . Consequently, using (31), a value of is chosen for the friction observer gain. The threshold velocity is chosen m/s, using a trial and error approach. equal to The parameters of both controllers are listed in Table II. Note that the tuned gain of the acceleration feedback controller is different from its initial value given by (27). Figs. 16(a) and 17(a) show that there is a slight lag (12 ms) in the responses, which is primarily due to the closed-loop
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Fig. 18.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 3, MAY 1998
Comparison of the position tracking error profiles.
Fig. 20. Comparison of the position tracking performance for a desired trajectory that approaches the motion limits.
Finally, in order to investigate the control performance when the actuation system exhibits increased nonlinear behavior, we present the experimental results with a trajectory that passes closer to the ends of the allowable motion domain. Fig. 20 compares the position responses using the two controllers. With the new desired trajectory, the piston reaches within 1.27 mm of the limits of the motion. A longer transition time has been used in this experiment, to prevent exceeding the maximum achievable speed in the desired trajectory; otherwise, saturation would occur. Once again, it is observed that the friction-compensating controller has a superior performance. VIII. CONCLUSIONS
Fig. 19.
Comparison of the velocity tracking error profiles.
dynamics of the system. As discussed earlier, ideally, the system response should have a time lag of 14 ms. Also, note that there are more high-frequency components in Fig. 17(c) than in Fig. 16(c), which is a result of including the acceleration error term in the control law. By shifting the position signals through 12 ms to compensate for the time lag, one can compare the tracking errors quantitatively. Fig. 18 shows the position tracking error signals. According to the rms values shown in this figure, using the friction-compensating controller results in reduction of the position error by a factor of 1.69. Likewise, Fig. 19 shows the velocity tracking error signals. The rms values show that using the friction-compensating controller results in reduction of the velocity error by a factor of 2.36.
Fast and accurate position control of a typical electrohydraulic manipulator, under friction nonlinearities, was considered in this paper. The manipulator considered is an integral part of an automated fish processing machine. In this machine, it is required to quickly move the object (cutter blade assembly) to its new position, as smoothly and as precisely as possible. Experiments showed that considerable static and dynamic friction exists in the prototype system. The nonlinear observer of Friedland and Park was used to estimate the dynamic friction. Convergence speed of the observer was studied, which was found to be useful for choosing the observer parameters. The Friedland–Park observer was combined with a first-order velocity filter, for real-time estimation of friction. A method was presented to convert the friction estimator to an acceleration observer. A novel control strategy for compensation of friction in electrohydraulic actuators was proposed that combines friction estimation with an acceleration feedback control law. The experimental results showed that, for a typical trajectory, the RMS position and velocity tracking errors were reduced by factors of 1.69 and 2.36, respectively. The approach pre-
TAFAZOLI et al.: TRACKING CONTROL OF AN ELECTROHYDRAULIC MANIPULATOR
sented in this paper is quite general and may be applied to any servomechanism with significant actuator dynamics and friction.
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[23] T. Matsuura, N. Shigematsu, K. Nakashima, and K. Moribe, “Hydraulic screwdown control system for mandrel mill,” IEEE Trans. Ind. Applicat., vol. 30, pp. 568–572, May/June 1994. [24] R. L. Andersson, A Robot Ping-Pong Player: Experiments in Real-Time Intelligent Control. Cambridge, MA: MIT Press, 1988, pp. 94–95.
ACKNOWLEDGMENT The authors wish to thank colleague Dr. E. A. Croft for her careful reading of the manuscript. REFERENCES [1] C. W. de Silva, “Research laboratory for fish processing automation,” Robot. Comput.-Integrated Manufacturing, vol. 9, no. 1, pp. 49–60, 1992. [2] S. Tafazoli, C. W. de Silva, and P. D. Lawrence, “Friction estimation in a planar electrohydraulic manipulator,” in Proc. Amer. Contr. Conf., Seattle, WA, June 1995, vol. 5, pp. 3294–3298. , “Position and force control of an electrohydraulic manipulator [3] in the presence of friction,” in Proc. IEEE Conf. Syst., Man, Cybern., Vancouver, BC, Oct. 1995, vol. 2, pp. 1687–1692. [4] H. E. Merritt, Hydraulic Control Systems. New York: Wiley, 1967. [5] B. Armstrong-H´elouvry, P. Dupont, and C. Canudas de Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol. 30, no. 7, pp. 1083–1138, May 1994. ˚ om, and P. Lischinsky, “A [6] C. Canudas de Wit, H. Olsson, K. J. Astr¨ new model for control of systems with friction,” IEEE Trans. Automat. Contr., vol. 40, pp. 419–425, Mar. 1995. [7] B. Armstrong-H´elouvry, Control of Machines with Friction. Boston, MA: Kluwer, 1991. [8] B. Friedland and Y. J. Park, “On adaptive friction compensation,” IEEE Trans. Automat. Contr., vol. 37, pp. 1609–1612, Oct. 1992. [9] B. Friedland and S. Mentzelopoulou, “On adaptive friction compensation without velocity measurement,” in Proc. IEEE Conf. Contr. Applicat., Dayton, OH, Sept. 1992, vol. 2, pp. 1076–1081. [10] S. Mentzelopoulou and B. Friedland, “Experimental evaluation of friction estimation and compensation techniques,” in Proc. Amer. Contr. Conf., Baltimore, MD, June 1994, vol. 3, pp. 3132–3136. [11] J. Amin, B. Friedland, and A. Harnoy, “Implementation of a friction estimation and compensation technique,” IEEE Contr. Syst. Mag., vol. 17, no. 4, pp. 71–76, Aug. 1997. [12] B. de Jager, “Acceleration assisted tracking control,” IEEE Contr. Syst. Mag., vol. 14, no. 5, pp. 20–27, Oct. 1994. [13] J. Studenny, P. R. B´elanger, and L. K. Daneshmand, “A digital implementation of the acceleration feedback control law on a PUMA 560 manipulator,” in Proc. 30th Conf. Decision Contr., Brighton, U.K., Dec. 1991, vol. 3, pp. 2639–2648. [14] D. Vischer and O. Khatib, “Design and development of highperformance torque-controlled joints,” IEEE Trans. Robot. Automat., vol. 11, no. 4, pp. 537–544, Aug. 1995. [15] L. E. Pfeffer, O. Khatib, and J. Hake, “Joint torque sensory feedback in the control of a PUMA manipulator,” IEEE Trans. Robot. Automat., vol. 5, pp. 418–425, Aug. 1989. [16] J. Y. S. Luh, W. D. Fisher, and R. P. C. Paul, “Joint torque control by a direct feedback for industrial robots,” IEEE Trans. Automat. Contr., vol. AC-28, pp. 153–161, Feb. 1983. [17] G. Vossoughi and M. Donath, “Dynamic feedback linearization for electrohydraulically actuated control systems,” J. Dynamic Syst., Measurement, Contr., Trans. ASME, vol. 117, pp. 468–477, Dec. 1995. [18] P. M. FitzSimons and J. J. Palazolo, “PART I: Modeling of a one-degreeof-freedom active hydraulic mount,” J. Dynamic Syst., Measurement, Contr., Trans. ASME, vol. 118, no. 3, pp. 439–442, Sept. 1996. [19] , “PART II: Control of a one-degree-of-freedom active hydraulic mount,” J. Dynamic Syst., Measurement, Contr., Trans. ASME, vol. 118, no. 3, pp. 443–448, Sept. 1996. [20] D. Karnopp, “Computer simulation of stick-slip friction in mechanical dynamic systems,” J. Dynamic Syst., Measurement, Contr., Trans. ASME, vol. 107, pp. 100–103, Mar. 1985. [21] C. W. de Silva, Control Sensors and Actuators. Englewood Cliffs, NJ: Prentice-Hall, 1989. [22] T. R. Welch, “The use of derivative pressure feedback in high performance hydraulic servomechanisms,” J. Eng. Ind., Trans. ASME, vol. 84, pp. 8–14, Feb. 1962.
Shahram Tafazoli (S’95–M’98) received the B.A.Sc. and M.A.Sc. (with honors) degrees in electrical engineering from Sharif University of Technology, Tehran, Iran, 1989 and 1991, respectively. He received the Ph.D. degree, also in electrical engineering, from the University of British Columbia (UBC), Vancouver, Canada, in January 1997. Currently, he is collaborating with the Robotics and Control Laboratory of UBC as a Research and Development Consultant. His research interests include friction estimation and compensation in mechanical systems, position and force control of robotic manipulators, and teleoperation. He has an application emphasis on teleoperated control and real-time payload estimation of the heavy-duty mobile hydraulic manipulators, such as excavators. Dr. Tafazoli held various fellowships and awards during his education and he was the Vice Chairman of the IEEE Control Systems Society, Vancouver Chapter, in 1995. He is also a member of the ASME.
Clarence W. de Silva (S’75–M’78–SM’85–F’98) received the Ph.D. degree in dynamic systems and control from the Massachusetts Institute of Technology, Cambridge, in 1978. Since 1988, he has been Professor of Mechanical Engineering and holder of the NSERC-BC Packers Chair in Industrial Automation at the University of British Columbia, Vancouver. He has authored 13 books and more than 100 journal papers, the most recent being Intelligent Control: Fuzzy Logic Applications (Boca Raton, FL: CRC, 1995). Dr. de Silva has been a Lilly Fellow, NASA/ASEE Fellow, Fullbright Fellow, ASI Fellow, Killam Fellow, and an ASME Fellow. He has served on editorial boards of 12 international journals including IEEE and ASME TRANSACTIONS, and is the Regional Editor, North America for the IFACaffiliated International Journal of Intelligent Real-Time Automation and Editor-in-Cheif of the International Journal of Knowledge-Based Intelligent Engineering Systems. He has received the Outstanding Contribution Award from the IEEE SMC Society, and the Meritorious Achievement Award from the Association of Professional Engineers of British Columbia.
Peter D. Lawrence (S’64–M’73) received the B.A.Sc. degree in electrical engineering from the University of Toronto, Canada, in 1965, the Masters degree in biomedical engineering from the University of Saskatchewan, Canada, in 1967, and the Ph.D. degree in computing and information science from Case Western Reserve University, Cleveland, OH, in 1970. He worked as a Guest Researcher at Chalmers University’s Applied Electronics Department in Goteborg, Sweden, between 1970 and 1972, and between 1972 and 1974 as a Research Staff Member and Lecturer in the Mechanical Engineering Department of the Massachusetts Institute of Technology, Cambridge. Since 1974, he has been with the University of British Columbia and is a Professor in the Department of Electrical and Computer Engineering. His main research interests include the application of real-time computing in the control interface between humans and machines, image processing, and mobile hydraulic machine modeling and control. Dr. Lawrence is a member of SAE and a registered Professional Engineer.
