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Friction compensation controller for load varying machine tool feed drive
International Journal of Machine Tools & Manufacture 96 (2015) 47–54
Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Friction compensation controller for load varying machine tool feed drive Wonkyun Lee a, Chan-Young Lee a, Young Hun Jeong b, Byung-Kwon Min a,n a b
Department of Mechanical Engineering, Yonsei University, Seoul 120-749, Republic of Korea School of Mechanical Engineering, Kyungpook National University, Daegu 702-701, Republic of Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 30 December 2014 Received in revised form 1 June 2015 Accepted 4 June 2015 Available online 11 June 2015
The load applied to a machine tool feed drive changes during the machining process as material is removed. This load change alters the Coulomb friction of the feed drive. Because Coulomb friction accounts for a large part of the total friction the friction compensation control accuracy of the feed drives is limited if this nonlinear change in the applied load is not considered. This paper presents a new friction compensation method that estimates the machine tool load in real time and considers its effect on friction characteristics. A friction observer based on a Kalman filter with load estimation is proposed for friction compensation control considering the applied load change. A specially designed feed drive testbed that enables the applied load to be modified easily was constructed for experimental verification. Control performance and friction estimation accuracy are demonstrated experimentally using the testbed. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Adaptive friction compensation Feed drive load Machine tool control Workpiece weight
1. Introduction Friction compensation control is one of the frequently used techniques to improve the motion control performance of machine tools [1,2]. Because the friction behavior of the machine tool feed drive is highly nonlinear [3,4] and time-variant during machining [5], implementation of friction compensation control is challenging. Friction compensation control methods can be categorized according to their use of friction models [1]. In model-free methods, such as robust control and adaptive control, the friction is considered as a part of a disturbance [6,7]. Yan et al. [8] proposed a combined two-degrees-of-freedom controller with a disturbance observer (DOB) to compensate for nonlinear friction, cogging effects and external disturbances. Lin et al. [9] presented a robust controller that combined a variable structure controller (VSC) with a DOB based disturbance compensator and compared the performance to a normal disturbance compensation controller. On the other hand, model-based methods exactly cancel out the effect of the friction force on the feed drive by adding an extra driving force equivalent to the estimated friction [10,11]. A friction model is incorporated into a feedback or feed-forward loop to estimate the friction. Although the friction effect can be effectively suppressed this method is difficult to apply to real machine tools because the performance of friction compensation is highly dependent on the accuracy of the friction model [12–14]. Coulomb n
Corresponding author. E-mail address: [email protected] (B.-K. Min).
http://dx.doi.org/10.1016/j.ijmachtools.2015.06.001 0890-6955/& 2015 Elsevier Ltd. All rights reserved.
friction that accounts for a large portion of the total friction of the feed drive is not constant because the load applied to the feed drive (mostly the weight of the workpiece) changes as material is removed or as different workpieces are loaded [15,16]. In a large machine tool tons of material may be removed during machining that reduces the friction force acting on the linear motion (LM) guide by as much as 30% of the initial condition. Therefore, precise estimation of the time-varying Coulomb friction force can improve the performance of the friction compensation controller. A variety of advanced methods have been proposed to overcome the performance degradation of model-based friction compensation caused by the inaccuracies of friction models [6,10,17]. In general, the DOB is used with the feed-forward friction compensator to predict the estimation error for friction ((measured friction) (estimated friction)) caused by model inaccuracies. However, the deterioration of the feed-forward friction compensator induced by tracking error and the time delay caused by the filter in the DOB can degrade the performance of friction compensation. Instead, we fundamentally compensated for the estimation error caused by model inaccuracies by correcting the model parameters instead of predicting and adding an estimation error to the observer output. We present a load estimator (LE) that estimates the change in weight of a workpiece without using additional sensors and introduce a novel friction compensation control algorithm that utilizes the estimated load. The proposed method is expected to be more robust than the aforementioned friction compensation methods that contain a feed-forward friction compensator and DOB, especially when the friction force is
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immediately changed (e.g., axis velocity reversal). In this paper we discuss the experimental results to demonstrate its performance. In Section 2, the effects of applied load on friction characteristics are investigated experimentally. An experimental setup and technique used to apply a variety of loads to the feed drive are proposed. Based on the results, a friction model of the feed drive is derived. Section 3 addresses the development of a state observer incorporating the load estimator. A friction compensation controller was designed and evaluated using simulation results. The experimental results are discussed in Section 4 and conclusions are drawn in Section 5.
Servo drive
dSPACE Feed drive
Charge amplifier
Host PC
2. Modeling of friction in feed drives Fig. 2. Experimental setup.
2.1. Effects of applied load on friction characteristics and To investigate the effects of applied load on friction characteristics a feed drive testbed is required so that the applied load can be controlled precisely. Because it is difficult to apply a variety of realistic loads to the feed drive table using weights a load control mechanism that does not require real weight was developed. Fig. 1 shows a schematic of the proposed feed drive with a designed load adjustment mechanism. Specially designed steel screws were used to push the LM blocks that were facing each other. The force applied to the LM block was controlled by the torque applied to the steel screw. Using the proposed mechanism loads up to 12.6 kN could be applied to the feed drive without placing a real weight on the table. The proposed mechanism does not change the moment of inertia of the feed drive, but, only the applied load. Consequently, the controlled feed drive dynamics are different from those when the load is applied by weights. However, the setup is sufficient for studying friction compensation because friction is not affected by the moment of inertia. In addition, all experiments were performed at constant velocity. A torque sensor and force sensor were installed to precisely measure the friction force of the ball screw and LM guide separately. The rotary torque sensor (4503A, Kistler) was installed between the servo motor and ball screw to measure the driving torque of the entire feed drive system (Tl ). The bi-directional force sensor (9217A, Kistler) was positioned between the ball screw nut and the table to measure the driving force acting on the table and LM guide (Fl ) only. The friction forces acting on the LM guide (FL ) and ball screw (FB ) were calculated using the measured force (Fl ) and torque (Tl ):
F L = Fl − ML x¨
(1)
FB = 2π Tl / pb − MB x¨ − Fl
(2)
where ML represents the mass of the table and the LM blocks. MB and pb denote the equivalent mass and lead of the ball screw, respectively. Fig. 2 shows the entire experimental setup. The feed drive was composed of a ball screw (BNK1205, THK) with a diameter of 12 mm and a lead of 5 mm connected to a permanent magnet synchronous motor (SGMAV, Yaskawa) and a servo drive (SGDV, Yaskawa) with a rated torque of 1.27 Nm. A set of LM guides (SSR20XW, THK) was used to support loads up to 78.4 kN. Ureabased grease (AFA, THK), with a kinematic viscosity of 20 mm2/s was applied to lubricate the ball screw and LM guides. The stroke of the feed drive was 150 mm. A real-time DSP controller (DS1103, dSPACE) was used for motion control and data acquisition. The resolution of the feed drive position used in the controller was 0.1 μm and the sampling frequency for the control was 10 kHz. The proportional and derivative gains of the PD controller were 50,000 and 100, respectively, where the position error was calculated in meters. The driving force acting on the table, u, was calculated as
u=
Vcommand × 0.1 × Trated × 2π L ballscrew
(3)
where Vcommand , Trated and L ballscrew represent the command signal generated in the controller, rated torque of the motor and ball screw lead, respectively. The load control mechanism including the torque wrench was calibrated using a load cell. The torque limit of the wrench with a ratchet head was set using a 1–10 scale dial. Fig. 3 shows the
Fig. 1. Schematic diagram of the experimental setup: mechanism for load application and sensor installation for friction measurements.
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49
Fig. 3. Experimental setup and results of the measurements of the relationship between the torque acting on the steel screw (ratchet step) and the normal force acting on the single linear motion (LM) block (WLA/4 ).
normal forces applied to the single LM block (WLA/4 ) by a steel screw using a load cell at each step of the torque dial. The force was measured 10 times for each torque step. The effect of the applied load on friction was investigated using the proposed experimental setup. The friction forces acting on ball screw and LM guide were measured as a function of change in the applied load for a feed rate of 10 mm/s. The experiments were repeated five times for each condition. As shown in Fig. 4, the friction acting on the LM guide increased proportionally to the applied load, while the effect of the applied load on the ball screw friction was negligible because the load acting on the table was fully supported by the LM blocks. Fig. 5 compares the Stribeck and hysteresis curves of the LM guide when the additional loads were applied. The Stribeck curves shifted on the y-axis in proportion to the applied load. This demonstrates that the effects of the applied load on the sliding parameters other than the Coulomb and breakaway friction are negligible. As shown in Fig. 5(b), only the peak values of the hysteresis curves increased and in proportion to the applied load due to variation in the Coulomb friction. This demonstrates that the effects of the applied load on the pre-sliding parameters were also negligible. 2.2. Friction Modeling The LuGre friction model was used to model the friction behavior of the feed drive as follows [18]:
⎛ |ẋ| ⎞ F (ẋ, z ) = σ0 z + σ1 ⎜⎜ẋ − σ0 z⎟ + μ v ẋ ̇ ⎟ f ⎝ n (x) ⎠
Fig. 5. Change in the (a) Stribeck and (b) hysteresis curves according to changes in load.
of the stiffness, damping, average deflection of the contacting bristles and viscous friction coefficient of the feed drive, respectively. fn (x ̇) is a function of the nonlinearity of the velocity–friction characteristics that is derived as
fn (ẋ) = FC + (F S − FC ) exp (− ẋ /vs )
(5)
where FC , FS and vs represent the Coulomb friction, break-away friction and Stribeck velocity, respectively. From Fig. 4, the Coulomb friction can be expressed as follows:
FC = FBC + μ LC (WLI + WLA )
(6)
where μ LC , WLI , WLA and FBC are the Coulomb friction coefficient, initial load including the preload, applied load acting on the LM blocks and the Coulomb friction of the ball screw, respectively. As shown in Fig. 5, the gap between the break-away friction and Coulomb friction is uniform regardless of the applied load. Therefore, this gap is identified to derive the break-away friction from the variable Coulomb friction.
3. Design and evaluation of the control system
(4)
3.1. Friction compensation controller design
where x, σ0 , σ1, z and μv denote the table position, effective values A control system consisting of a proportional-derivative (PD) controller and an observer-based friction compensator was designed for friction compensation control. A state observer incorporating a load estimator was used to estimate the friction under variation in the applied load. The feed drive model was updated for each sampling using the applied load estimated by the load estimator. A Kalman filter was used as a state observer to minimize the effect of noise in the input signal. To design a state observer a simple dynamic model of the feed drive is proposed as
MS x¨ + BS ẋ = u − F (ẋ, z )
(7)
where MS and B S represent the equivalent mass and damping of the system, respectively. The equivalent mass includes the mass of the table and weight of the workpiece, as well as the mass moment of inertia of the motor and ball screw. The model can be expressed in a state space form and is given as follows:
Ẋ = A¯ X + B¯ u, Fig. 4. Friction force acting on the (a) ball screw and (b) LM guide.
where
⎡ x⎤ X = ⎢ x ̇ ⎥, ⎢⎣ ⎥⎦ z
Y = [1 0 0] X (8)
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⎡0 ⎢ ⎢ ¯ A = ⎢0 − ⎢ ⎢ ⎢⎣0
1 σ 1 + μv + B S MS
⎤ ⎥
0 −
⎛ ẋ ⎞ ⎟⎥ σ 0 ⎜1 − σ 1 fn (ẋ) ⎠ ⎝ ⎥
⎥ ⎥ ⎥⎦
MS
1
− σ0 f
ẋ
n (x)̇
From the estimated state variables the friction force can be calculated using the following equation.
⎡0⎤ ⎢1⎥ ¯ B = ⎢ MS ⎥ . ⎢ ⎥ ⎣0⎦
,
⎤ ⎡ ⎛ x ̇ ⎞⎥ ^ ^ ⎟⎟ X F = ⎢0 σ1 + μ v σ0 ⎜⎜1 − σ1 ⎢⎣ fn (ẋ) ⎠⎥⎦ ⎝
The dynamic model of the feed drive derived in Eq. (8) is discretized as
X[k + 1] = A d X[k] + Bd u[k] + w[k], X[k]
⎡ x (k )⎤ ⎢ ⎥ = ⎢ ẋ (k )⎥, ⎢⎣ z (k ) ⎥⎦
The accuracy of the estimated friction value depends on the accuracy of the feed drive model. If the model is not sufficiently accurate the state observer becomes unstable because the system noise in Eq. (9), w[k], has a non-zero mean value. With the assumption that there is no observer error at the k-th sampling time Eq. (9) can be arranged as follows by combining Eqs. (10) and (11):
^ ^ X[k + 1 k + 1] − X[k + 1 k] = K s C w[k]
Y[k] = CX[k] + v[k]
(9)
where w[k] and v[k] represent the zero mean white noise of the process and measurement at the k-th sampling with covariance Q and R, respectively. The state observer can be expressed as
⎡ ⎤ ^ ^ ^ X[k + 1 k + 1] = X[k + 1 k] + K s ⎣⎢Y[k + 1] − CX[k + 1 k] ⎦⎥
(10)
(
^ ^ ^ ^ WLA [k + 1] = WLA [k] + sign (ẋ (k ) ) K w F[k + 1 k + 1] − F[k + 1 k]
e[k + 1 k] = A d e[k k] − w[k]
(12)
and the error covariance is defined using the observer error as
Ω = E ⎡⎣e e T ⎤⎦
(13)
The observer becomes stable when the error covariance matrix
Ω saturates to zero [19]. Therefore, the observer gain (Ks ) is derived as follows. The error covariance matrix at the (kþ 1)th sampling time estimated at the k-th sampling time is given by Ω[k + 1 k] = E ⎡⎣e[k + 1 k] e[k + 1 k]T ⎤⎦ ⎡ ⎤ = E ⎣A d e[k k]e[k k]T A d T − A d e[k k]w k T − w k e[k k]T A d T + w[k]w[k] T ⎦ = A d Ω[k k]A d T + Q
(14)
Therefore, the error covariance matrix at the (k þ1)th sampling time is updated using the measured value Y[k þ 1] as
Ω[k + 1 k + 1] = E ⎡⎣e[k + 1 | k + 1] e[k + 1 | k + 1]T ⎤⎦ ⎡ = E ⎣⎢ (I − K s C ) e[k + 1 | k] − K s v[k + 1] )
(
(I − K s C )T
(20)
(11)
^ X[k + 1 k] is the state variable at the (k þ1)th sampling time estimated at the k-th sampling time before the measured value Y[k + 1] is updated. The state variables are updated to X^[k + 1 k + 1] using the measured value given by Eq. (10). The observer error is the difference between the estimated and actual state that is expressed as
T [k + 1 | k ]
)
where K w denotes the load estimator gain.
^ ^ X[k + 1 k] = A d X[k k] + Bd u[k]
(e
(19)
According to Eq. (19), the effects of load on the feed drive are proportional to the difference between the estimated state variables before and after updating Eq. (10). Therefore, the load estimator is designed to minimize the difference between F^[k + 1 k] and ^ F[k + 1 k + 1] such that
where
^ e[k k] = X[k k] − X[k],
(18)
)
⎤ − v[k + 1]T K s T ⎥⎦
)
T
= (I − K s C ) Ω[k + 1 | k](I − K s C ) + K s RK s
3.2. Evaluation of the proposed friction compensation control method To demonstrate the performance of the proposed method the tracking errors of feed drives controlled by five controllers were simulated and compared: (1) a conventional PD controller, (2) VSC with the DOB based disturbance compensator (DOBVSC) [9], (3) PD controller with the feed-forward friction compensator with the DOB (PD þFFþ DOB) [6,10,17], (4) PD controller with the state observer based friction compensator (PD þFC) [20] and (5) proposed controller (PD þ FC þLE). Fig. 6 shows the block diagram of each controller. As shown in Fig. 7(a) and (b) an additional load was applied from 0 to 10 kN during triangular wave motion. The same proportional and derivative gains were used for each controller. As shown in Fig. 7(c) the tracking errors were significantly lower after applying the friction compensation control methods. The tracking errors in the region between 2 and 2.2 s demonstrate that the proposed method is more effective than the DOBVSC and PDþ FFþDOB when the axis velocity reversal occurs because of the time delay caused by the filter in the DOB. The difference between the tracking errors of the feed drive controlled by the model-based friction compensation methods (PD þFC and PDþ FC þLE) in the region between 4 and 4.2 s demonstrates that the proposed load estimator maintained the friction compensation performance even when the applied load was changed. The peak and RMS values of the tracking errors in this region are listed in Table 1.
4. Results and discussion T
(15)
4.1. Parameter identification for the feed drive model
For zero saturation of the error covariance
∂Ω = − 2 (I − K s C ) Ω C T + 2 K s R = 0 ∂K s
(16)
Consequently, the observer is stable when the observer gain Ks is set as −1 K s = ΩC T ⎡⎣CΩC T + R⎤⎦
(17)
All of the feed drive model parameters were determined using the experimental results. Equivalent mass and damping of the system were identified using the step response of the feed drive. The Coulomb friction of the ball screw (FBC ), the Coulomb friction coefficient of the LM guide (μ LC ) and the load initially applied to the LM guide (WLI ) were derived and the values are listed in Table. 2. The friction model parameters with the exception of those for Coulomb friction were divided into sliding friction parameters (FS ,
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51
Fig. 7. Simulation conditions and results: (a) position profile, (b) applied load and (c) comparison of tracking errors. Fig. 6. Block diagrams of the friction (disturbance) compensation controllers: (a) PD, (b) DOBVSC, (c) PD þ FFþ DOB, (d) PD þFC and (e) PD þ FCþ LE.
vs and μv ) and presliding parameters (σ0 and σ1), and then identified individually using a genetic algorithm-based method [20]. Stribeck and friction-position hysteresis curves were obtained experimentally and used to identify the sliding and pre-sliding parameters, respectively. All parameters were identified separately according to the direction of motion to improve the model accuracy. Fig. 8 compares the experimentally measured friction characteristics and their estimations using the friction model with identified parameters. Table. 3 lists the experimentally determined feed drive model parameters.
Table 1 Peak and RMS values of the tracking errors. Control methods
PD DOBVSC PD þ FFþDOB PD þ FC PD þ FCþLE
Tracking error (μm) Peak
RMS
12.0 2.4 1.2 0.7 0.7
10.2 0.9 0.6 0.3 0.2
4.2. Evaluation of the friction compensation control algorithm Before verifying the performance of the proposed friction compensation controller under load varying conditions the state observer based friction compensation controller (PD þFC) was evaluated experimentally without applying the additional load (only the table weight was acting as load). When the system underwent reciprocal motion, as illustrated in Fig. 9(a), the proposed state observer accurately estimated the friction acting on the LM guide and the ball screw with an accuracy of 93% or better as shown in Fig. 9(b) and (c). Fig. 9(d) compares the tracking errors of the feed drive controlled by the different compensation control
Table 2 Identified Coulomb friction parameters. Symbols (units)
Values Positive dir.
Negative dir.
FBC (N) μ LC
63.43
64.95
3.66 10 4
3.66 10 4
WLI (kN)
15.2
15.2
52
W. Lee et al. / International Journal of Machine Tools & Manufacture 96 (2015) 47–54
Fig. 8. Friction characteristics of the experimental setup and identified feed drive model: (a) Stribeck and (b) hysteresis curves. Table 3 Identified feed drive model parameters. Symbols (units)
Values Positive dir.
Negative dir.
MS (kg) B S (N s/m) FS (N) vs (m/s) μv (N s/m)
100
100
380
260
FC þ 30.00 0.0010 1826.5
FC þ 22.98 0.0014 1673.5
σ0 (N/m) σ1 (N s/m)
3.35 107 4.54 103
3.35 107 4.54 103
methods (e.g., DOBVSC, PDþ FFþDOB, PD þFC). The method based on the state observer was more effective than the others, especially when the axis velocity reversal occurs. 4.3. Friction compensation results under load varying conditions Fig. 10 shows the experimental results evaluating the performance of the proposed friction compensation controller (PD þFC þ LE). Fig. 10 (a)–(c) and Fig. 10 (d)–(f) show the results for an applied load of 10.9 kN (Step 9 in Fig. 3) and 12.6 kN (Step 10 in Fig. 3), respectively. Fig. 10 (a) and (d) compares the friction estimated by the state observer with and without the load estimator. Only the friction acting on the LM guide was compared because the friction change at the ball screw was negligible. The estimation accuracy of the friction improved when the load estimator was implemented. The RMS friction estimation errors were reduced from 3.08 to 1.18 N when a load of 10.9 kN was applied and from 4.95 to 1.05 N when a load of 12.6 kN was applied. Fig. 10 (b) and
Fig. 9. Experimental results of friction compensation without application of an additional load: (a) reference position profile, friction force acting on the (b) LM guide and (c) ball screw estimated by the proposed friction observer and (d) the tracking errors of feed drive controlled by various friction-compensation controllers.
(e) compares the tracking errors of the feed drive controlled by each controller. As presented in Section 3.2 the performance of the PDþ FC was degraded due to the applied load while the other methods (e.g., DOBVSC and PDþ FFþDOB) maintained their performances. However, this performance degradation was reduced after applying the load estimator (PD þFC þ LE). Table 4 shows the RMS values of the estimation error, tracking error and estimated load, using the state observer based friction compensator with and without a load estimator. These results demonstrate that the proposed load estimator improved the tracking performance of the load varying feed drive.
5. Conclusions This paper presents a friction compensation controller for a
W. Lee et al. / International Journal of Machine Tools & Manufacture 96 (2015) 47–54
53
Fig. 10. Experimental results for verifying the load estimator (LE) for (a–c) a 10.9 kN load and (d–f) a 12.6 kN applied load. (a) and (d) are friction forces acting on the LM guide, (b) and (e) are tracking errors and (c) and (f) are applied loads.
Table 4 Comparison of the estimation error, tracking error, and the estimated load. Applied load (kN)
10.9 12.6
Estimation error (N, RMS)
Tracking error (μm, RMS)
Estimated load (kN, RMS)
PD þ FC
PD þFCþ LE
PD þ FC
PDþ FCþ LE
PD þFC þLE
3.08 4.95
1.18 1.05
1.65 2.18
0.98 1.10
10.9 13.1
machine tool feed drive to deal with the varying load caused by material removal and its effects on friction characteristic change. A friction compensation controller based on the Kalman filter that incorporated a load estimator was proposed and verified by comparing the performance with a variety of friction compensation control methods. A ball screw type feed drive testbed that enabled the applied load to be modified was constructed to evaluate the proposed controller. The applied load to the feed drive was estimated using the proposed load estimator with an accuracy of 96% or better. The performance degradation of the state observer based friction compensator (PD þFC) caused by the applied load was compensated by using the load estimator (PD þFC þ LE). The experimental results demonstrated the effectiveness of the proposed friction compensation control method under changing loads during machining, particularly for large variation in load.
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[9]
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Acknowledgments [12]
This research was supported by the government-funded research program of the Korea Institute of Machinery and Materials, Republic of Korea, ‘Mechatronics Optimization of High Speed & High Accuracy Machinery Equipment.’
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Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Friction compensation controller for load varying machine tool feed drive Wonkyun Lee a, Chan-Young Lee a, Young Hun Jeong b, Byung-Kwon Min a,n a b
Department of Mechanical Engineering, Yonsei University, Seoul 120-749, Republic of Korea School of Mechanical Engineering, Kyungpook National University, Daegu 702-701, Republic of Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 30 December 2014 Received in revised form 1 June 2015 Accepted 4 June 2015 Available online 11 June 2015
The load applied to a machine tool feed drive changes during the machining process as material is removed. This load change alters the Coulomb friction of the feed drive. Because Coulomb friction accounts for a large part of the total friction the friction compensation control accuracy of the feed drives is limited if this nonlinear change in the applied load is not considered. This paper presents a new friction compensation method that estimates the machine tool load in real time and considers its effect on friction characteristics. A friction observer based on a Kalman filter with load estimation is proposed for friction compensation control considering the applied load change. A specially designed feed drive testbed that enables the applied load to be modified easily was constructed for experimental verification. Control performance and friction estimation accuracy are demonstrated experimentally using the testbed. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Adaptive friction compensation Feed drive load Machine tool control Workpiece weight
1. Introduction Friction compensation control is one of the frequently used techniques to improve the motion control performance of machine tools [1,2]. Because the friction behavior of the machine tool feed drive is highly nonlinear [3,4] and time-variant during machining [5], implementation of friction compensation control is challenging. Friction compensation control methods can be categorized according to their use of friction models [1]. In model-free methods, such as robust control and adaptive control, the friction is considered as a part of a disturbance [6,7]. Yan et al. [8] proposed a combined two-degrees-of-freedom controller with a disturbance observer (DOB) to compensate for nonlinear friction, cogging effects and external disturbances. Lin et al. [9] presented a robust controller that combined a variable structure controller (VSC) with a DOB based disturbance compensator and compared the performance to a normal disturbance compensation controller. On the other hand, model-based methods exactly cancel out the effect of the friction force on the feed drive by adding an extra driving force equivalent to the estimated friction [10,11]. A friction model is incorporated into a feedback or feed-forward loop to estimate the friction. Although the friction effect can be effectively suppressed this method is difficult to apply to real machine tools because the performance of friction compensation is highly dependent on the accuracy of the friction model [12–14]. Coulomb n
Corresponding author. E-mail address: [email protected] (B.-K. Min).
http://dx.doi.org/10.1016/j.ijmachtools.2015.06.001 0890-6955/& 2015 Elsevier Ltd. All rights reserved.
friction that accounts for a large portion of the total friction of the feed drive is not constant because the load applied to the feed drive (mostly the weight of the workpiece) changes as material is removed or as different workpieces are loaded [15,16]. In a large machine tool tons of material may be removed during machining that reduces the friction force acting on the linear motion (LM) guide by as much as 30% of the initial condition. Therefore, precise estimation of the time-varying Coulomb friction force can improve the performance of the friction compensation controller. A variety of advanced methods have been proposed to overcome the performance degradation of model-based friction compensation caused by the inaccuracies of friction models [6,10,17]. In general, the DOB is used with the feed-forward friction compensator to predict the estimation error for friction ((measured friction) (estimated friction)) caused by model inaccuracies. However, the deterioration of the feed-forward friction compensator induced by tracking error and the time delay caused by the filter in the DOB can degrade the performance of friction compensation. Instead, we fundamentally compensated for the estimation error caused by model inaccuracies by correcting the model parameters instead of predicting and adding an estimation error to the observer output. We present a load estimator (LE) that estimates the change in weight of a workpiece without using additional sensors and introduce a novel friction compensation control algorithm that utilizes the estimated load. The proposed method is expected to be more robust than the aforementioned friction compensation methods that contain a feed-forward friction compensator and DOB, especially when the friction force is
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immediately changed (e.g., axis velocity reversal). In this paper we discuss the experimental results to demonstrate its performance. In Section 2, the effects of applied load on friction characteristics are investigated experimentally. An experimental setup and technique used to apply a variety of loads to the feed drive are proposed. Based on the results, a friction model of the feed drive is derived. Section 3 addresses the development of a state observer incorporating the load estimator. A friction compensation controller was designed and evaluated using simulation results. The experimental results are discussed in Section 4 and conclusions are drawn in Section 5.
Servo drive
dSPACE Feed drive
Charge amplifier
Host PC
2. Modeling of friction in feed drives Fig. 2. Experimental setup.
2.1. Effects of applied load on friction characteristics and To investigate the effects of applied load on friction characteristics a feed drive testbed is required so that the applied load can be controlled precisely. Because it is difficult to apply a variety of realistic loads to the feed drive table using weights a load control mechanism that does not require real weight was developed. Fig. 1 shows a schematic of the proposed feed drive with a designed load adjustment mechanism. Specially designed steel screws were used to push the LM blocks that were facing each other. The force applied to the LM block was controlled by the torque applied to the steel screw. Using the proposed mechanism loads up to 12.6 kN could be applied to the feed drive without placing a real weight on the table. The proposed mechanism does not change the moment of inertia of the feed drive, but, only the applied load. Consequently, the controlled feed drive dynamics are different from those when the load is applied by weights. However, the setup is sufficient for studying friction compensation because friction is not affected by the moment of inertia. In addition, all experiments were performed at constant velocity. A torque sensor and force sensor were installed to precisely measure the friction force of the ball screw and LM guide separately. The rotary torque sensor (4503A, Kistler) was installed between the servo motor and ball screw to measure the driving torque of the entire feed drive system (Tl ). The bi-directional force sensor (9217A, Kistler) was positioned between the ball screw nut and the table to measure the driving force acting on the table and LM guide (Fl ) only. The friction forces acting on the LM guide (FL ) and ball screw (FB ) were calculated using the measured force (Fl ) and torque (Tl ):
F L = Fl − ML x¨
(1)
FB = 2π Tl / pb − MB x¨ − Fl
(2)
where ML represents the mass of the table and the LM blocks. MB and pb denote the equivalent mass and lead of the ball screw, respectively. Fig. 2 shows the entire experimental setup. The feed drive was composed of a ball screw (BNK1205, THK) with a diameter of 12 mm and a lead of 5 mm connected to a permanent magnet synchronous motor (SGMAV, Yaskawa) and a servo drive (SGDV, Yaskawa) with a rated torque of 1.27 Nm. A set of LM guides (SSR20XW, THK) was used to support loads up to 78.4 kN. Ureabased grease (AFA, THK), with a kinematic viscosity of 20 mm2/s was applied to lubricate the ball screw and LM guides. The stroke of the feed drive was 150 mm. A real-time DSP controller (DS1103, dSPACE) was used for motion control and data acquisition. The resolution of the feed drive position used in the controller was 0.1 μm and the sampling frequency for the control was 10 kHz. The proportional and derivative gains of the PD controller were 50,000 and 100, respectively, where the position error was calculated in meters. The driving force acting on the table, u, was calculated as
u=
Vcommand × 0.1 × Trated × 2π L ballscrew
(3)
where Vcommand , Trated and L ballscrew represent the command signal generated in the controller, rated torque of the motor and ball screw lead, respectively. The load control mechanism including the torque wrench was calibrated using a load cell. The torque limit of the wrench with a ratchet head was set using a 1–10 scale dial. Fig. 3 shows the
Fig. 1. Schematic diagram of the experimental setup: mechanism for load application and sensor installation for friction measurements.
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49
Fig. 3. Experimental setup and results of the measurements of the relationship between the torque acting on the steel screw (ratchet step) and the normal force acting on the single linear motion (LM) block (WLA/4 ).
normal forces applied to the single LM block (WLA/4 ) by a steel screw using a load cell at each step of the torque dial. The force was measured 10 times for each torque step. The effect of the applied load on friction was investigated using the proposed experimental setup. The friction forces acting on ball screw and LM guide were measured as a function of change in the applied load for a feed rate of 10 mm/s. The experiments were repeated five times for each condition. As shown in Fig. 4, the friction acting on the LM guide increased proportionally to the applied load, while the effect of the applied load on the ball screw friction was negligible because the load acting on the table was fully supported by the LM blocks. Fig. 5 compares the Stribeck and hysteresis curves of the LM guide when the additional loads were applied. The Stribeck curves shifted on the y-axis in proportion to the applied load. This demonstrates that the effects of the applied load on the sliding parameters other than the Coulomb and breakaway friction are negligible. As shown in Fig. 5(b), only the peak values of the hysteresis curves increased and in proportion to the applied load due to variation in the Coulomb friction. This demonstrates that the effects of the applied load on the pre-sliding parameters were also negligible. 2.2. Friction Modeling The LuGre friction model was used to model the friction behavior of the feed drive as follows [18]:
⎛ |ẋ| ⎞ F (ẋ, z ) = σ0 z + σ1 ⎜⎜ẋ − σ0 z⎟ + μ v ẋ ̇ ⎟ f ⎝ n (x) ⎠
Fig. 5. Change in the (a) Stribeck and (b) hysteresis curves according to changes in load.
of the stiffness, damping, average deflection of the contacting bristles and viscous friction coefficient of the feed drive, respectively. fn (x ̇) is a function of the nonlinearity of the velocity–friction characteristics that is derived as
fn (ẋ) = FC + (F S − FC ) exp (− ẋ /vs )
(5)
where FC , FS and vs represent the Coulomb friction, break-away friction and Stribeck velocity, respectively. From Fig. 4, the Coulomb friction can be expressed as follows:
FC = FBC + μ LC (WLI + WLA )
(6)
where μ LC , WLI , WLA and FBC are the Coulomb friction coefficient, initial load including the preload, applied load acting on the LM blocks and the Coulomb friction of the ball screw, respectively. As shown in Fig. 5, the gap between the break-away friction and Coulomb friction is uniform regardless of the applied load. Therefore, this gap is identified to derive the break-away friction from the variable Coulomb friction.
3. Design and evaluation of the control system
(4)
3.1. Friction compensation controller design
where x, σ0 , σ1, z and μv denote the table position, effective values A control system consisting of a proportional-derivative (PD) controller and an observer-based friction compensator was designed for friction compensation control. A state observer incorporating a load estimator was used to estimate the friction under variation in the applied load. The feed drive model was updated for each sampling using the applied load estimated by the load estimator. A Kalman filter was used as a state observer to minimize the effect of noise in the input signal. To design a state observer a simple dynamic model of the feed drive is proposed as
MS x¨ + BS ẋ = u − F (ẋ, z )
(7)
where MS and B S represent the equivalent mass and damping of the system, respectively. The equivalent mass includes the mass of the table and weight of the workpiece, as well as the mass moment of inertia of the motor and ball screw. The model can be expressed in a state space form and is given as follows:
Ẋ = A¯ X + B¯ u, Fig. 4. Friction force acting on the (a) ball screw and (b) LM guide.
where
⎡ x⎤ X = ⎢ x ̇ ⎥, ⎢⎣ ⎥⎦ z
Y = [1 0 0] X (8)
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⎡0 ⎢ ⎢ ¯ A = ⎢0 − ⎢ ⎢ ⎢⎣0
1 σ 1 + μv + B S MS
⎤ ⎥
0 −
⎛ ẋ ⎞ ⎟⎥ σ 0 ⎜1 − σ 1 fn (ẋ) ⎠ ⎝ ⎥
⎥ ⎥ ⎥⎦
MS
1
− σ0 f
ẋ
n (x)̇
From the estimated state variables the friction force can be calculated using the following equation.
⎡0⎤ ⎢1⎥ ¯ B = ⎢ MS ⎥ . ⎢ ⎥ ⎣0⎦
,
⎤ ⎡ ⎛ x ̇ ⎞⎥ ^ ^ ⎟⎟ X F = ⎢0 σ1 + μ v σ0 ⎜⎜1 − σ1 ⎢⎣ fn (ẋ) ⎠⎥⎦ ⎝
The dynamic model of the feed drive derived in Eq. (8) is discretized as
X[k + 1] = A d X[k] + Bd u[k] + w[k], X[k]
⎡ x (k )⎤ ⎢ ⎥ = ⎢ ẋ (k )⎥, ⎢⎣ z (k ) ⎥⎦
The accuracy of the estimated friction value depends on the accuracy of the feed drive model. If the model is not sufficiently accurate the state observer becomes unstable because the system noise in Eq. (9), w[k], has a non-zero mean value. With the assumption that there is no observer error at the k-th sampling time Eq. (9) can be arranged as follows by combining Eqs. (10) and (11):
^ ^ X[k + 1 k + 1] − X[k + 1 k] = K s C w[k]
Y[k] = CX[k] + v[k]
(9)
where w[k] and v[k] represent the zero mean white noise of the process and measurement at the k-th sampling with covariance Q and R, respectively. The state observer can be expressed as
⎡ ⎤ ^ ^ ^ X[k + 1 k + 1] = X[k + 1 k] + K s ⎣⎢Y[k + 1] − CX[k + 1 k] ⎦⎥
(10)
(
^ ^ ^ ^ WLA [k + 1] = WLA [k] + sign (ẋ (k ) ) K w F[k + 1 k + 1] − F[k + 1 k]
e[k + 1 k] = A d e[k k] − w[k]
(12)
and the error covariance is defined using the observer error as
Ω = E ⎡⎣e e T ⎤⎦
(13)
The observer becomes stable when the error covariance matrix
Ω saturates to zero [19]. Therefore, the observer gain (Ks ) is derived as follows. The error covariance matrix at the (kþ 1)th sampling time estimated at the k-th sampling time is given by Ω[k + 1 k] = E ⎡⎣e[k + 1 k] e[k + 1 k]T ⎤⎦ ⎡ ⎤ = E ⎣A d e[k k]e[k k]T A d T − A d e[k k]w k T − w k e[k k]T A d T + w[k]w[k] T ⎦ = A d Ω[k k]A d T + Q
(14)
Therefore, the error covariance matrix at the (k þ1)th sampling time is updated using the measured value Y[k þ 1] as
Ω[k + 1 k + 1] = E ⎡⎣e[k + 1 | k + 1] e[k + 1 | k + 1]T ⎤⎦ ⎡ = E ⎣⎢ (I − K s C ) e[k + 1 | k] − K s v[k + 1] )
(
(I − K s C )T
(20)
(11)
^ X[k + 1 k] is the state variable at the (k þ1)th sampling time estimated at the k-th sampling time before the measured value Y[k + 1] is updated. The state variables are updated to X^[k + 1 k + 1] using the measured value given by Eq. (10). The observer error is the difference between the estimated and actual state that is expressed as
T [k + 1 | k ]
)
where K w denotes the load estimator gain.
^ ^ X[k + 1 k] = A d X[k k] + Bd u[k]
(e
(19)
According to Eq. (19), the effects of load on the feed drive are proportional to the difference between the estimated state variables before and after updating Eq. (10). Therefore, the load estimator is designed to minimize the difference between F^[k + 1 k] and ^ F[k + 1 k + 1] such that
where
^ e[k k] = X[k k] − X[k],
(18)
)
⎤ − v[k + 1]T K s T ⎥⎦
)
T
= (I − K s C ) Ω[k + 1 | k](I − K s C ) + K s RK s
3.2. Evaluation of the proposed friction compensation control method To demonstrate the performance of the proposed method the tracking errors of feed drives controlled by five controllers were simulated and compared: (1) a conventional PD controller, (2) VSC with the DOB based disturbance compensator (DOBVSC) [9], (3) PD controller with the feed-forward friction compensator with the DOB (PD þFFþ DOB) [6,10,17], (4) PD controller with the state observer based friction compensator (PD þFC) [20] and (5) proposed controller (PD þ FC þLE). Fig. 6 shows the block diagram of each controller. As shown in Fig. 7(a) and (b) an additional load was applied from 0 to 10 kN during triangular wave motion. The same proportional and derivative gains were used for each controller. As shown in Fig. 7(c) the tracking errors were significantly lower after applying the friction compensation control methods. The tracking errors in the region between 2 and 2.2 s demonstrate that the proposed method is more effective than the DOBVSC and PDþ FFþDOB when the axis velocity reversal occurs because of the time delay caused by the filter in the DOB. The difference between the tracking errors of the feed drive controlled by the model-based friction compensation methods (PD þFC and PDþ FC þLE) in the region between 4 and 4.2 s demonstrates that the proposed load estimator maintained the friction compensation performance even when the applied load was changed. The peak and RMS values of the tracking errors in this region are listed in Table 1.
4. Results and discussion T
(15)
4.1. Parameter identification for the feed drive model
For zero saturation of the error covariance
∂Ω = − 2 (I − K s C ) Ω C T + 2 K s R = 0 ∂K s
(16)
Consequently, the observer is stable when the observer gain Ks is set as −1 K s = ΩC T ⎡⎣CΩC T + R⎤⎦
(17)
All of the feed drive model parameters were determined using the experimental results. Equivalent mass and damping of the system were identified using the step response of the feed drive. The Coulomb friction of the ball screw (FBC ), the Coulomb friction coefficient of the LM guide (μ LC ) and the load initially applied to the LM guide (WLI ) were derived and the values are listed in Table. 2. The friction model parameters with the exception of those for Coulomb friction were divided into sliding friction parameters (FS ,
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Fig. 7. Simulation conditions and results: (a) position profile, (b) applied load and (c) comparison of tracking errors. Fig. 6. Block diagrams of the friction (disturbance) compensation controllers: (a) PD, (b) DOBVSC, (c) PD þ FFþ DOB, (d) PD þFC and (e) PD þ FCþ LE.
vs and μv ) and presliding parameters (σ0 and σ1), and then identified individually using a genetic algorithm-based method [20]. Stribeck and friction-position hysteresis curves were obtained experimentally and used to identify the sliding and pre-sliding parameters, respectively. All parameters were identified separately according to the direction of motion to improve the model accuracy. Fig. 8 compares the experimentally measured friction characteristics and their estimations using the friction model with identified parameters. Table. 3 lists the experimentally determined feed drive model parameters.
Table 1 Peak and RMS values of the tracking errors. Control methods
PD DOBVSC PD þ FFþDOB PD þ FC PD þ FCþLE
Tracking error (μm) Peak
RMS
12.0 2.4 1.2 0.7 0.7
10.2 0.9 0.6 0.3 0.2
4.2. Evaluation of the friction compensation control algorithm Before verifying the performance of the proposed friction compensation controller under load varying conditions the state observer based friction compensation controller (PD þFC) was evaluated experimentally without applying the additional load (only the table weight was acting as load). When the system underwent reciprocal motion, as illustrated in Fig. 9(a), the proposed state observer accurately estimated the friction acting on the LM guide and the ball screw with an accuracy of 93% or better as shown in Fig. 9(b) and (c). Fig. 9(d) compares the tracking errors of the feed drive controlled by the different compensation control
Table 2 Identified Coulomb friction parameters. Symbols (units)
Values Positive dir.
Negative dir.
FBC (N) μ LC
63.43
64.95
3.66 10 4
3.66 10 4
WLI (kN)
15.2
15.2
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W. Lee et al. / International Journal of Machine Tools & Manufacture 96 (2015) 47–54
Fig. 8. Friction characteristics of the experimental setup and identified feed drive model: (a) Stribeck and (b) hysteresis curves. Table 3 Identified feed drive model parameters. Symbols (units)
Values Positive dir.
Negative dir.
MS (kg) B S (N s/m) FS (N) vs (m/s) μv (N s/m)
100
100
380
260
FC þ 30.00 0.0010 1826.5
FC þ 22.98 0.0014 1673.5
σ0 (N/m) σ1 (N s/m)
3.35 107 4.54 103
3.35 107 4.54 103
methods (e.g., DOBVSC, PDþ FFþDOB, PD þFC). The method based on the state observer was more effective than the others, especially when the axis velocity reversal occurs. 4.3. Friction compensation results under load varying conditions Fig. 10 shows the experimental results evaluating the performance of the proposed friction compensation controller (PD þFC þ LE). Fig. 10 (a)–(c) and Fig. 10 (d)–(f) show the results for an applied load of 10.9 kN (Step 9 in Fig. 3) and 12.6 kN (Step 10 in Fig. 3), respectively. Fig. 10 (a) and (d) compares the friction estimated by the state observer with and without the load estimator. Only the friction acting on the LM guide was compared because the friction change at the ball screw was negligible. The estimation accuracy of the friction improved when the load estimator was implemented. The RMS friction estimation errors were reduced from 3.08 to 1.18 N when a load of 10.9 kN was applied and from 4.95 to 1.05 N when a load of 12.6 kN was applied. Fig. 10 (b) and
Fig. 9. Experimental results of friction compensation without application of an additional load: (a) reference position profile, friction force acting on the (b) LM guide and (c) ball screw estimated by the proposed friction observer and (d) the tracking errors of feed drive controlled by various friction-compensation controllers.
(e) compares the tracking errors of the feed drive controlled by each controller. As presented in Section 3.2 the performance of the PDþ FC was degraded due to the applied load while the other methods (e.g., DOBVSC and PDþ FFþDOB) maintained their performances. However, this performance degradation was reduced after applying the load estimator (PD þFC þ LE). Table 4 shows the RMS values of the estimation error, tracking error and estimated load, using the state observer based friction compensator with and without a load estimator. These results demonstrate that the proposed load estimator improved the tracking performance of the load varying feed drive.
5. Conclusions This paper presents a friction compensation controller for a
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53
Fig. 10. Experimental results for verifying the load estimator (LE) for (a–c) a 10.9 kN load and (d–f) a 12.6 kN applied load. (a) and (d) are friction forces acting on the LM guide, (b) and (e) are tracking errors and (c) and (f) are applied loads.
Table 4 Comparison of the estimation error, tracking error, and the estimated load. Applied load (kN)
10.9 12.6
Estimation error (N, RMS)
Tracking error (μm, RMS)
Estimated load (kN, RMS)
PD þ FC
PD þFCþ LE
PD þ FC
PDþ FCþ LE
PD þFC þLE
3.08 4.95
1.18 1.05
1.65 2.18
0.98 1.10
10.9 13.1
machine tool feed drive to deal with the varying load caused by material removal and its effects on friction characteristic change. A friction compensation controller based on the Kalman filter that incorporated a load estimator was proposed and verified by comparing the performance with a variety of friction compensation control methods. A ball screw type feed drive testbed that enabled the applied load to be modified was constructed to evaluate the proposed controller. The applied load to the feed drive was estimated using the proposed load estimator with an accuracy of 96% or better. The performance degradation of the state observer based friction compensator (PD þFC) caused by the applied load was compensated by using the load estimator (PD þFC þ LE). The experimental results demonstrated the effectiveness of the proposed friction compensation control method under changing loads during machining, particularly for large variation in load.
[4] [5]
[6] [7]
[8]
[9]
[10] [11]
Acknowledgments [12]
This research was supported by the government-funded research program of the Korea Institute of Machinery and Materials, Republic of Korea, ‘Mechatronics Optimization of High Speed & High Accuracy Machinery Equipment.’
[13] [14]
[15]
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