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Control of ball screw drives based on disturbance response optimization
CIRP Annals - Manufacturing Technology 62 (2013) 387–390
Contents lists available at SciVerse ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Control of ball screw drives based on disturbance response optimization Kaan Erkorkmaz (2)*, Yasin Hosseinkhani Precision Controls Laboratory, Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, Canada
A R T I C L E I N F O
A B S T R A C T
Keywords: Control Drive Vibration
This paper presents a new method for designing control laws for ball screw drives by directly optimizing the load side disturbance response against cutting forces. The design applies, concurrently, the principles of pole placement and loop shaping, and is easy to implement in practice. In addition to good low frequency disturbance rejection, the control law provides active vibration damping, which reduces the magnification of tracking errors near the drive’s mechanical resonance. Effectiveness of the proposed strategy is demonstrated in machining and high speed tracking experiments, where its performance is compared to the industry standard P-PI cascade control law. ß 2013 CIRP.
1. Introduction The principal function of machine tool drives is to realize the desired feed motion between the tool and workpiece as accurately as possible, while ensuring that disturbances, such as process forces and nonlinear friction, have the least detrimental impact on the dynamic positioning accuracy. As the general trend in feed drive design continues towards increasing the velocity and acceleration capabilities, and the closed-loop bandwidth, it is inevitable that the control law has to deal with mechanical resonance(s) which may lie within, or very close to, its responsive frequency range [1]. This has motivated research in developing feedforward type command shaping techniques, to avoid exciting the drive’s structure by reference inputs [1,2]; and feedback based vibration damping methods, which conquer the oscillatory response of the mechanical structure and are key to improving the disturbance rejection of the drive around resonances. Some of the recently proposed solutions, which achieve active damping, rely on employing: acceleration feedback [1,3–5]; additional frictional actuators [6]; a state observer [5]; H1 robust control [4]; sliding mode control [7,8]; and generalized predictive control [9]. With the considerations that: additional hardware can be costly to integrate; and for acceptance by industry, a feed drive controller needs to be robust and easy to implement, a pole placement solution was proposed in [10]. Its structure assumes the form of parallel PD and PID filters, and its implementation does not require additional sensors, actuators, or the use of advanced level control theory (except for computing the feedback gains). This structure was demonstrated to tolerate up to 40% drive stiffness variation and %150 load mass increase. In this paper, we present a new design approach which builds upon the pole placement structure in [10], but achieves the two following improvements:
* Corresponding author. 0007-8506/$ – see front matter ß 2013 CIRP. http://dx.doi.org/10.1016/j.cirp.2013.03.138
1. The design directly targets minimization of the load side disturbance response against cutting forces. This improves the ball screw drive’s disturbance rejection across the frequency spectrum, rather than only making a local improvement at the expense of worsening the dynamic stiffness at another frequency. 2. The phase delay contributed by the power electronics is considered and compensated, by applying loop shaping principles in conjunction with pole placement control. This helps to improve the worst case dynamic stiffness by 33%, over that achievable with the earlier design in [10]. 2. Controller design There have been numerous contributions dedicated to modeling of ball screw drives. While high order and distributed parameter models have successfully been proposed, in many cases a simple lumped model like the one in Fig. 1 can be sufficient for characterizing the dominant control and disturbance response [1]. This model is intuitive and easy to implement in industry. When long stroke is concerned, modal parameters may shift as a function of table position, due to change in the effective drive stiffness k. Then, gain scheduling can be applied to retain model accuracy [4]. To keep the presentation simple, only disturbances acting on the load side (i.e. d) are shown in Fig. 1. Disturbances acting on the motor (m1) side, such, as rotary bearing friction, can be considered simply by injecting them through the same channel as the control input (u).
Fig. 1. Simplified ball screw model capturing first vibration mode.
K. Erkorkmaz, Y. Hosseinkhani / CIRP Annals - Manufacturing Technology 62 (2013) 387–390
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Fig. 3. Proposed control scheme using pole placement and loop shaping.
The primary objective is to minimize |Gdist(s)| as much as possible, both in peak value and also overall shape, which helps to diminish the worst-case dynamic compliance. Fig. 2. Modeled and measured open-loop frequency response functions.
Loop transfer function
The experimental setup used is a precision ball screw with 20 mm diameter and 350 mm stroke. Stiffness variation due to travel is less than 10%, hence gain scheduling is not used. A 3 kW servo motor provides the actuation. Rotational and translational positions (x1, x2) are measured through encoders which provide measurement resolutions equivalent to 10 nm and 20 nm of table motion, respectively. A dSPACE system is used for implementing the control algorithms at a sampling frequency of 20 kHz. The response of the ball screw drive to control (u) and disturbance (d) inputs was measured by applying sinusoidal current commands and impact hammer tests. The measurements are shown in Fig. 2. Using the model in Fig. 1b, the control signal equivalent drive parameters were identified as m1 = 1.858 103 V/(rad/s2), m2 = 0.379 103 V/(rad/s2), b1 = 1.020 103 V/(rad/s), b2 = 0 V/ (rad/s) (due to aerostatic guideways), k = 247.1 V/rad, and c = 0.0222 V/(rad/s). Conversion of these parameters to SI units can be realized by considering the current amplifier gain (1.7193 A/V), motor torque constant (0.57 Nm/A), and lead screw pitch (20 mm). The high pitch value leads to a low gearing ratio between the motor and table. While this facilitates faster feed motion and higher table acceleration, it also renders the ball screw mechanism more vulnerable against cutting and friction disturbances acting on the load side. Another important parameter is the loop delay Tdelay = 0.333 ms, caused by the 3.9 kHz pulse–width modulation (PWM) in the current amplifier and additional computational and sampling delays in the system. Frequency response functions (FRF’s) predicted by the model have been overlaid on top of the measurements in Fig. 2. As seen, the model, which captures the first vibration mode at 141 Hz with 4% damping, is able to replicate both the controlled input and table disturbance responses quite well; up to 250–300 Hz. In transfer matrix form, the drive model can be written as: uðsÞ G1 ðsÞ Gd1 ðsÞ uðsÞ x1 ðsÞ ¼ ¼ ½ GðsÞ Gd ðsÞ (1) x2 ðsÞ G2 ðsÞ Gd2 ðsÞ dðdÞ dðdÞ The objective is to find a feedback controller KðsÞ ¼ ½ K 1 ðsÞ K 2 ðsÞ which optimizes the load side closed loop disturbance response against cutting forces for a wide frequency range. Merely boosting loop crossover frequency can sometimes deteriorate the disturbance response near the vibration mode. On the other hand, two controllers designed for different frequency ranges, one for vibration damping and the other for rigid body motion, may interfere with each other when the desired closed loop bandwidth is near the vibration mode. Hence, in this work direct minimization of the closed loop disturbance response is targeted, with full attention to all terms in the feedback control law. The design and tuning steps are carried out while inspecting the following two transfer functions. Closed-loop disturbance response
Gdist ðsÞ ¼ S21 Gd1 þ S22 Gd2 ; S ¼
S11 S21
S12 S22
¼ ðI þ GKÞ1
(2)
Although the drive has two outputs, by applying block manipulation the analysis can be transformed into SISO form [10], allowing L(s) to be found as: LðsÞ ¼ K 1 G1 þ K 2 G2
(3)
The stability margins, crossover frequency, and minimum distance to ‘1’ are inspected from Bode and Nyquist plots of L(s). Minimum margins of GM 1.5 and PM 258 are targeted. The proposed control scheme is shown in Fig. 3. It consists of jointly applying pole placement and loop shaping principles. The function and design of each component is explained in the proceeding paragraphs. For comparison, a P-PI cascade controller was also designed, where the velocity loop is closed with rotational feedback and the position loop with translational feedback. P-PI cascade represents the industry mainstream. This controller was designed according to the guidelines in [11], and also possesses velocity and acceleration feedforward terms. 2.1. Pole placement controller (PPC) The position and velocity states z ¼ ½ x1 x2 x˙ 1 x˙ 2 T of the dynamic model in Fig. 1b are either measurable or can be estimated by numerical differentiation (due to high encoder resolution). This eliminates the need to use an observer, which can be problematic in terms of modeling errors and loop delays. The integrated table position is introduced as a fifth state Rt (x2i ðtÞ ¼ 0 x2 ðt Þdt ), to boost low frequency disturbance rejection on the load side by enforcing integral action. This also helps compensate for the motion loss in the preloaded nut [12]. The PPC assumes a simple PD-PID form: K x1r x1 u p pc ¼ K x1 þ K v1 s; K x2 þ i2 þ K v2 s (4) x2r x2 s |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} K p pc ðsÞ
The feedback gains Kx1, K v1 , Kx2, Ki2, K v2 are computed to achieve the desired closed loop poles [13]. The oscillatory pole pair at 141 Hz is re-assigned by shifting their real component further to the left in the s-plane (Fig. 4), following the concept of low authority LQG vibration control [14]. This accelerates the exponential decay without altering the mechanical structure’s damped frequency, which would be costly in terms of control effort. With help of the lead filter (explained next), b could be increased up to 5.5, targeting nearly 5 improvement in the damping. This dramatically reduces the resonance at 141 Hz in the load disturbance response, as seen in Fig. 5 for ‘PPC + LS’. The remaining poles were tuned, while inspecting Gdist and L, to be: ) p3 ¼ l l ¼ 100 Hz qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5) where : v ¼ 100 Hz; z ¼ 2:0 p4;5 ¼ zv v z2 1 It should be noted that while the disturbance response was optimized considering the model transfer function, the actual disturbance responses in Fig. 5, measured through impact hammer testing, displayed two inconsistencies with their models. The first one is the mode at 43 Hz, which is due to pitching motion of the
K. Erkorkmaz, Y. Hosseinkhani / CIRP Annals - Manufacturing Technology 62 (2013) 387–390
389
Fig. 4. Active damping of structural vibrations by pole placement.
table pivoted around the guideway blocks. On the setup, the ball screw and guideways are mounted at the same level. Hence, it is very difficult to excite or control this mode from the motor, unless a mechanical design change is made. However, when an impact is delivered to the top of the table and acceleration is measured at the same level, which is how the measurements were taken, this mode shows up nearly unaltered regardless of the feedback control law used. The second inconsistency is in the actual resonance magnitudes. It is believed that due to motion loss in the nut (in the order of 1–2 mm), the nut interface does not function like an ideal spring. Hence, the damping action generated by the motor cannot be fully transferred to the table. This leaves a portion of the vibration unquenched. Nevertheless, the model and experimental results in Fig. 5 show that improving the theoretical response also improves the actual disturbance rejection. This is further validated in the machining results presented in Section 3.
The 0.333 ms phase delay, contributed by sampling and PWM, poses a serious limitation on the achievable closed loop bandwidth. This is overcome by careful design of a lead filter D(s), which lifts up the loop phase around the cross over frequency vc (determined to be 230 Hz, as seen in Fig. 6): KðT d s þ 1Þ 1=T d ¼ 55 Hz; 1=T a ¼ 880 Hz where : K ¼ 0:1667 Tas þ 1
used sparingly. Hence, N(s) was designed as: NðsÞ ¼
4 Y s2 þ 2zm vm s þ v2m s2 þ 2zD vm s þ v2m m¼1
(7)
Above, v1 = 280 Hz, v2 = 420 Hz, v3 = 1080 Hz, v4 = 1750 Hz, z1 = 0.40, z2 = 0.15, z3 = 0.03, z4 = 0.10, zD = 0.707. In the case of PPC + LS, traces of the modes at 1080 Hz and 1750 Hz can still be seen in Figs. 6 and 7. However, since at these frequencies, L(jv) is distant from ‘1’, these modes are not problematic. 2.4. Command generator
2.2. Lead filter
DðsÞ ¼
Fig. 6. Loop transfer function magnitude for PPC + LS and P-PI schemes.
(6)
This filter provides 61.98 phase lead around vc. By examining the loop shape (Fig. 7), its gain contribution around vc was set to 0.7, in order to obtain adequate stability margins. The lead filter dramatically improves the specifications and disturbance rejection achievable with PPC. Without it, PPC tuning could only be increased up to b = 4, l = v = 50 Hz and z = 0.707. 2.3. Notch filters The challenge in damping a low frequency mode is ensuring that higher frequency modes are not excited in a way which can cause instability. On the setup, such modes originate mainly from rotational dynamics, which are little influenced by table position and can be suppressed with a notch filter pack N(s). Spikes in the input sensitivity function S(s) = 1/(1 + L(s)) provide excellent indicators of frequencies that require notching. In order not to induce too much phase loss near the crossover, the notch effect is
The reference trajectory represents the desired motion for x2. The rotational position command has to be adjusted to account for the anticipated elastic deformation of the drive. Hence, a command generator is used in the form: m2 s2 þ ðb2 þ cÞs þ k Gr1 ; Gr2 ¼ 1 Gr ðsÞ ¼ (8) ; where : Gr1 ¼ Gr2 cs þ k Gr1 is obtained by considering the free body diagram for m2 in Fig. 1b, when no disturbance is active. 2.5. Model inverse and trajectory prefilter To improve command following, an inverse of the drive model is used in feedforward. Also, a trajectory prefilter is employed to correct for distortions in the tracking transfer function (x2/xr) caused by discretization and causal implementation of the controller terms: G f ¼ 1 þ K vel Gl p f s þ K acc G2l p f s2 þ K jerk G3l p f s3 þ K sna p G4l p f s4
(9)
Above, Glpf is a first order filter with 80 Hz cut-off frequency. K vel , Kacc, Kjerk, Ksnap are tuned automatically from a single tracking experiment, so that correlations of the velocity, acceleration, jerk, and snap commands in the tracking error profile are minimized [10]. 3. Experimental results The disturbance response optimized controller has been validated in machining and high speed tracking experiments.
Fig. 5. Load side disturbance response for open- and closed-loop cases.
Fig. 7. Nyquist diagrams for PPC + loop shaping and P-PI control.
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K. Erkorkmaz, Y. Hosseinkhani / CIRP Annals - Manufacturing Technology 62 (2013) 387–390
Fig. 10. High speed tracking (vel: 0.5 m/s, acc: 8.2 m/s2, jerk: 200 m/s3).
Fig. 8. Experimental setup.
The ball screw drive was mounted on the x-axis of an OKK Vertical Machining Center 410, equipped with a 3500 rpm spindle (Fig. 8). Cutting tests were performed by slotting Aluminum 6065 with a 25.4 mm diameter 2 flute helical end mill at a chip load of 0.254 mm/tooth. The spindle speed was varied in steps of 600, 1200, 1800, 2400, 3000, 3500 rpm, and the feed motion was provided by the ball screw drive. Experimental tracking errors, measured from the linear encoder, are shown in Fig. 9. For comparison, results for the P-PI controller are also presented. To ensure the repeatability of the results, each cut was performed three times. Considering Fig. 9, PPC + LS provides much more uniform disturbance rejection as the frequency content of the cutting force varies. In addition, disturbance rejection of PPC + LS is, in general, much better than that of P-PI cascade control, particularly around the resonance frequency of the drive. These
observed trends are consistent with the disturbance response graphs in Fig. 5. Considering Fig. 10, the proposed design also provides superior tracking over P-PI cascade control. In implementing both controllers, neither friction nor lead error compensation was used. Since PPC + LS has higher loop gain, it also has a tendency to amplify the effect of lead errors further. These errors, which can be considered as ‘output’ type disturbances [7], are typically smaller than the errors caused by cutting forces. However, depending on the application of a machine tool, a different compromise may need to be struck between sensitivity to cutting forces versus lead errors and sensor noise. Then, the design procedure outlined in this paper can be applied accordingly. 4. Conclusions This paper has presented a new method for designing feed drive controllers, by directly optimizing the cutting force disturbance response of the electro-mechanical system. The guidelines are straightforward and intuitive. The new control law displays superior rejection of cutting force disturbances compared to the established P-PI cascade architecture. The tracking performance is also very good. Future work targets investigating and augmenting the robustness of this control law. Acknowledgments The authors gratefully acknowledge the financial support of NSERC and CFI, equipment contributions from Heidenhein, NSK, Omron, and New Way Precision, and the assistance of University of Waterloo CNC technician, Mr. Robert Wagner. References
Fig. 9. Table positioning errors during machining.
[1] Altintas Y, Verl A, Brecher C, Uriarte L, Pritschow G (2011) Machine Tool Feed Drives. Annals of the CIRP 60(2):779–796. [2] Altintas Y, Khoshdarregi MR (2012) Contour Error Control of CNC Machine Tools with Vibration Avoidance. Annals of the CIRP 61(1):335–338. [3] Chen Y-C, Tlusty J (1995) Effect of Low-Friction Guideways and Lead-Screw Flexibility on Dynamics of High-Speed Machines. Annals of the CIRP 44(1):353– 356. [4] Symens W, Van Brussel H, Swevers J (2004) Gain-Scheduling Control of Machine Tools with Varying Structural Flexibility. Annals of the CIRP 53(1):321–324. ˜ a I, Illarramendi A (2005) New Control Techni[5] Zatarain M, Ruiz De Argandon ques Based on State Space Observers for Improving the Precision and Dynamic Behaviour of Machine Tools. Annals of the CIRP 54(1):393–396. [6] Verl A, Frey S (2012) Improvement of Feed Drive Dynamics by Means of SemiActive Damping. Annals of the CIRP 61(1):351–354. [7] Kamalzadeh A, Erkorkmaz K (2007) Compensation of Axial Vibrations in Ball Screw Drives. Annals of the CIRP 56(1):373–378. [8] Okwudire CE, Altintas Y (2009) Minimum-Tracking Error Control of Flexible Ballscrew Drives using a Discrete-Time Sliding Mode Controller. ASME Journal of Dynamic Systems Measurement and Control 131(5). 051006.1–051006.12. [9] Dumur D, Susanu M, Aubourg M (2008) Complex Form Machining with Axis Drive Predictive Control. Annals of the CIRP 57(1):399–402. [10] Gordon, D.J., Erkorkmaz, K., Accurate Control of Ball Screw Drives using PolePlacement Vibration Damping and a Novel Trajectory Prefilter, Precision Engineering, in press. [11] Ellis GH (2004) Control System Design Guide, 3rd ed. Elsevier Academic Press, New York, USA. [12] Cuttino JF, Dow TA, Knight BF (1997) Analytical and Experimental Identification of Nonlinearities in a Single-Nut Preloaded Ball Screw. ASME Journal of Mechanical Design 119:15–19. [13] Ogata K (2002) Modern Control Engineering, Prentice Hall, New Jersey, USA. [14] Gawronski WK (1998) Dynamics and Control of Structures: A Modal Approach, Springer, New York, USA.
Contents lists available at SciVerse ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Control of ball screw drives based on disturbance response optimization Kaan Erkorkmaz (2)*, Yasin Hosseinkhani Precision Controls Laboratory, Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, Canada
A R T I C L E I N F O
A B S T R A C T
Keywords: Control Drive Vibration
This paper presents a new method for designing control laws for ball screw drives by directly optimizing the load side disturbance response against cutting forces. The design applies, concurrently, the principles of pole placement and loop shaping, and is easy to implement in practice. In addition to good low frequency disturbance rejection, the control law provides active vibration damping, which reduces the magnification of tracking errors near the drive’s mechanical resonance. Effectiveness of the proposed strategy is demonstrated in machining and high speed tracking experiments, where its performance is compared to the industry standard P-PI cascade control law. ß 2013 CIRP.
1. Introduction The principal function of machine tool drives is to realize the desired feed motion between the tool and workpiece as accurately as possible, while ensuring that disturbances, such as process forces and nonlinear friction, have the least detrimental impact on the dynamic positioning accuracy. As the general trend in feed drive design continues towards increasing the velocity and acceleration capabilities, and the closed-loop bandwidth, it is inevitable that the control law has to deal with mechanical resonance(s) which may lie within, or very close to, its responsive frequency range [1]. This has motivated research in developing feedforward type command shaping techniques, to avoid exciting the drive’s structure by reference inputs [1,2]; and feedback based vibration damping methods, which conquer the oscillatory response of the mechanical structure and are key to improving the disturbance rejection of the drive around resonances. Some of the recently proposed solutions, which achieve active damping, rely on employing: acceleration feedback [1,3–5]; additional frictional actuators [6]; a state observer [5]; H1 robust control [4]; sliding mode control [7,8]; and generalized predictive control [9]. With the considerations that: additional hardware can be costly to integrate; and for acceptance by industry, a feed drive controller needs to be robust and easy to implement, a pole placement solution was proposed in [10]. Its structure assumes the form of parallel PD and PID filters, and its implementation does not require additional sensors, actuators, or the use of advanced level control theory (except for computing the feedback gains). This structure was demonstrated to tolerate up to 40% drive stiffness variation and %150 load mass increase. In this paper, we present a new design approach which builds upon the pole placement structure in [10], but achieves the two following improvements:
* Corresponding author. 0007-8506/$ – see front matter ß 2013 CIRP. http://dx.doi.org/10.1016/j.cirp.2013.03.138
1. The design directly targets minimization of the load side disturbance response against cutting forces. This improves the ball screw drive’s disturbance rejection across the frequency spectrum, rather than only making a local improvement at the expense of worsening the dynamic stiffness at another frequency. 2. The phase delay contributed by the power electronics is considered and compensated, by applying loop shaping principles in conjunction with pole placement control. This helps to improve the worst case dynamic stiffness by 33%, over that achievable with the earlier design in [10]. 2. Controller design There have been numerous contributions dedicated to modeling of ball screw drives. While high order and distributed parameter models have successfully been proposed, in many cases a simple lumped model like the one in Fig. 1 can be sufficient for characterizing the dominant control and disturbance response [1]. This model is intuitive and easy to implement in industry. When long stroke is concerned, modal parameters may shift as a function of table position, due to change in the effective drive stiffness k. Then, gain scheduling can be applied to retain model accuracy [4]. To keep the presentation simple, only disturbances acting on the load side (i.e. d) are shown in Fig. 1. Disturbances acting on the motor (m1) side, such, as rotary bearing friction, can be considered simply by injecting them through the same channel as the control input (u).
Fig. 1. Simplified ball screw model capturing first vibration mode.
K. Erkorkmaz, Y. Hosseinkhani / CIRP Annals - Manufacturing Technology 62 (2013) 387–390
388
Fig. 3. Proposed control scheme using pole placement and loop shaping.
The primary objective is to minimize |Gdist(s)| as much as possible, both in peak value and also overall shape, which helps to diminish the worst-case dynamic compliance. Fig. 2. Modeled and measured open-loop frequency response functions.
Loop transfer function
The experimental setup used is a precision ball screw with 20 mm diameter and 350 mm stroke. Stiffness variation due to travel is less than 10%, hence gain scheduling is not used. A 3 kW servo motor provides the actuation. Rotational and translational positions (x1, x2) are measured through encoders which provide measurement resolutions equivalent to 10 nm and 20 nm of table motion, respectively. A dSPACE system is used for implementing the control algorithms at a sampling frequency of 20 kHz. The response of the ball screw drive to control (u) and disturbance (d) inputs was measured by applying sinusoidal current commands and impact hammer tests. The measurements are shown in Fig. 2. Using the model in Fig. 1b, the control signal equivalent drive parameters were identified as m1 = 1.858 103 V/(rad/s2), m2 = 0.379 103 V/(rad/s2), b1 = 1.020 103 V/(rad/s), b2 = 0 V/ (rad/s) (due to aerostatic guideways), k = 247.1 V/rad, and c = 0.0222 V/(rad/s). Conversion of these parameters to SI units can be realized by considering the current amplifier gain (1.7193 A/V), motor torque constant (0.57 Nm/A), and lead screw pitch (20 mm). The high pitch value leads to a low gearing ratio between the motor and table. While this facilitates faster feed motion and higher table acceleration, it also renders the ball screw mechanism more vulnerable against cutting and friction disturbances acting on the load side. Another important parameter is the loop delay Tdelay = 0.333 ms, caused by the 3.9 kHz pulse–width modulation (PWM) in the current amplifier and additional computational and sampling delays in the system. Frequency response functions (FRF’s) predicted by the model have been overlaid on top of the measurements in Fig. 2. As seen, the model, which captures the first vibration mode at 141 Hz with 4% damping, is able to replicate both the controlled input and table disturbance responses quite well; up to 250–300 Hz. In transfer matrix form, the drive model can be written as: uðsÞ G1 ðsÞ Gd1 ðsÞ uðsÞ x1 ðsÞ ¼ ¼ ½ GðsÞ Gd ðsÞ (1) x2 ðsÞ G2 ðsÞ Gd2 ðsÞ dðdÞ dðdÞ The objective is to find a feedback controller KðsÞ ¼ ½ K 1 ðsÞ K 2 ðsÞ which optimizes the load side closed loop disturbance response against cutting forces for a wide frequency range. Merely boosting loop crossover frequency can sometimes deteriorate the disturbance response near the vibration mode. On the other hand, two controllers designed for different frequency ranges, one for vibration damping and the other for rigid body motion, may interfere with each other when the desired closed loop bandwidth is near the vibration mode. Hence, in this work direct minimization of the closed loop disturbance response is targeted, with full attention to all terms in the feedback control law. The design and tuning steps are carried out while inspecting the following two transfer functions. Closed-loop disturbance response
Gdist ðsÞ ¼ S21 Gd1 þ S22 Gd2 ; S ¼
S11 S21
S12 S22
¼ ðI þ GKÞ1
(2)
Although the drive has two outputs, by applying block manipulation the analysis can be transformed into SISO form [10], allowing L(s) to be found as: LðsÞ ¼ K 1 G1 þ K 2 G2
(3)
The stability margins, crossover frequency, and minimum distance to ‘1’ are inspected from Bode and Nyquist plots of L(s). Minimum margins of GM 1.5 and PM 258 are targeted. The proposed control scheme is shown in Fig. 3. It consists of jointly applying pole placement and loop shaping principles. The function and design of each component is explained in the proceeding paragraphs. For comparison, a P-PI cascade controller was also designed, where the velocity loop is closed with rotational feedback and the position loop with translational feedback. P-PI cascade represents the industry mainstream. This controller was designed according to the guidelines in [11], and also possesses velocity and acceleration feedforward terms. 2.1. Pole placement controller (PPC) The position and velocity states z ¼ ½ x1 x2 x˙ 1 x˙ 2 T of the dynamic model in Fig. 1b are either measurable or can be estimated by numerical differentiation (due to high encoder resolution). This eliminates the need to use an observer, which can be problematic in terms of modeling errors and loop delays. The integrated table position is introduced as a fifth state Rt (x2i ðtÞ ¼ 0 x2 ðt Þdt ), to boost low frequency disturbance rejection on the load side by enforcing integral action. This also helps compensate for the motion loss in the preloaded nut [12]. The PPC assumes a simple PD-PID form: K x1r x1 u p pc ¼ K x1 þ K v1 s; K x2 þ i2 þ K v2 s (4) x2r x2 s |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} K p pc ðsÞ
The feedback gains Kx1, K v1 , Kx2, Ki2, K v2 are computed to achieve the desired closed loop poles [13]. The oscillatory pole pair at 141 Hz is re-assigned by shifting their real component further to the left in the s-plane (Fig. 4), following the concept of low authority LQG vibration control [14]. This accelerates the exponential decay without altering the mechanical structure’s damped frequency, which would be costly in terms of control effort. With help of the lead filter (explained next), b could be increased up to 5.5, targeting nearly 5 improvement in the damping. This dramatically reduces the resonance at 141 Hz in the load disturbance response, as seen in Fig. 5 for ‘PPC + LS’. The remaining poles were tuned, while inspecting Gdist and L, to be: ) p3 ¼ l l ¼ 100 Hz qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5) where : v ¼ 100 Hz; z ¼ 2:0 p4;5 ¼ zv v z2 1 It should be noted that while the disturbance response was optimized considering the model transfer function, the actual disturbance responses in Fig. 5, measured through impact hammer testing, displayed two inconsistencies with their models. The first one is the mode at 43 Hz, which is due to pitching motion of the
K. Erkorkmaz, Y. Hosseinkhani / CIRP Annals - Manufacturing Technology 62 (2013) 387–390
389
Fig. 4. Active damping of structural vibrations by pole placement.
table pivoted around the guideway blocks. On the setup, the ball screw and guideways are mounted at the same level. Hence, it is very difficult to excite or control this mode from the motor, unless a mechanical design change is made. However, when an impact is delivered to the top of the table and acceleration is measured at the same level, which is how the measurements were taken, this mode shows up nearly unaltered regardless of the feedback control law used. The second inconsistency is in the actual resonance magnitudes. It is believed that due to motion loss in the nut (in the order of 1–2 mm), the nut interface does not function like an ideal spring. Hence, the damping action generated by the motor cannot be fully transferred to the table. This leaves a portion of the vibration unquenched. Nevertheless, the model and experimental results in Fig. 5 show that improving the theoretical response also improves the actual disturbance rejection. This is further validated in the machining results presented in Section 3.
The 0.333 ms phase delay, contributed by sampling and PWM, poses a serious limitation on the achievable closed loop bandwidth. This is overcome by careful design of a lead filter D(s), which lifts up the loop phase around the cross over frequency vc (determined to be 230 Hz, as seen in Fig. 6): KðT d s þ 1Þ 1=T d ¼ 55 Hz; 1=T a ¼ 880 Hz where : K ¼ 0:1667 Tas þ 1
used sparingly. Hence, N(s) was designed as: NðsÞ ¼
4 Y s2 þ 2zm vm s þ v2m s2 þ 2zD vm s þ v2m m¼1
(7)
Above, v1 = 280 Hz, v2 = 420 Hz, v3 = 1080 Hz, v4 = 1750 Hz, z1 = 0.40, z2 = 0.15, z3 = 0.03, z4 = 0.10, zD = 0.707. In the case of PPC + LS, traces of the modes at 1080 Hz and 1750 Hz can still be seen in Figs. 6 and 7. However, since at these frequencies, L(jv) is distant from ‘1’, these modes are not problematic. 2.4. Command generator
2.2. Lead filter
DðsÞ ¼
Fig. 6. Loop transfer function magnitude for PPC + LS and P-PI schemes.
(6)
This filter provides 61.98 phase lead around vc. By examining the loop shape (Fig. 7), its gain contribution around vc was set to 0.7, in order to obtain adequate stability margins. The lead filter dramatically improves the specifications and disturbance rejection achievable with PPC. Without it, PPC tuning could only be increased up to b = 4, l = v = 50 Hz and z = 0.707. 2.3. Notch filters The challenge in damping a low frequency mode is ensuring that higher frequency modes are not excited in a way which can cause instability. On the setup, such modes originate mainly from rotational dynamics, which are little influenced by table position and can be suppressed with a notch filter pack N(s). Spikes in the input sensitivity function S(s) = 1/(1 + L(s)) provide excellent indicators of frequencies that require notching. In order not to induce too much phase loss near the crossover, the notch effect is
The reference trajectory represents the desired motion for x2. The rotational position command has to be adjusted to account for the anticipated elastic deformation of the drive. Hence, a command generator is used in the form: m2 s2 þ ðb2 þ cÞs þ k Gr1 ; Gr2 ¼ 1 Gr ðsÞ ¼ (8) ; where : Gr1 ¼ Gr2 cs þ k Gr1 is obtained by considering the free body diagram for m2 in Fig. 1b, when no disturbance is active. 2.5. Model inverse and trajectory prefilter To improve command following, an inverse of the drive model is used in feedforward. Also, a trajectory prefilter is employed to correct for distortions in the tracking transfer function (x2/xr) caused by discretization and causal implementation of the controller terms: G f ¼ 1 þ K vel Gl p f s þ K acc G2l p f s2 þ K jerk G3l p f s3 þ K sna p G4l p f s4
(9)
Above, Glpf is a first order filter with 80 Hz cut-off frequency. K vel , Kacc, Kjerk, Ksnap are tuned automatically from a single tracking experiment, so that correlations of the velocity, acceleration, jerk, and snap commands in the tracking error profile are minimized [10]. 3. Experimental results The disturbance response optimized controller has been validated in machining and high speed tracking experiments.
Fig. 5. Load side disturbance response for open- and closed-loop cases.
Fig. 7. Nyquist diagrams for PPC + loop shaping and P-PI control.
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K. Erkorkmaz, Y. Hosseinkhani / CIRP Annals - Manufacturing Technology 62 (2013) 387–390
Fig. 10. High speed tracking (vel: 0.5 m/s, acc: 8.2 m/s2, jerk: 200 m/s3).
Fig. 8. Experimental setup.
The ball screw drive was mounted on the x-axis of an OKK Vertical Machining Center 410, equipped with a 3500 rpm spindle (Fig. 8). Cutting tests were performed by slotting Aluminum 6065 with a 25.4 mm diameter 2 flute helical end mill at a chip load of 0.254 mm/tooth. The spindle speed was varied in steps of 600, 1200, 1800, 2400, 3000, 3500 rpm, and the feed motion was provided by the ball screw drive. Experimental tracking errors, measured from the linear encoder, are shown in Fig. 9. For comparison, results for the P-PI controller are also presented. To ensure the repeatability of the results, each cut was performed three times. Considering Fig. 9, PPC + LS provides much more uniform disturbance rejection as the frequency content of the cutting force varies. In addition, disturbance rejection of PPC + LS is, in general, much better than that of P-PI cascade control, particularly around the resonance frequency of the drive. These
observed trends are consistent with the disturbance response graphs in Fig. 5. Considering Fig. 10, the proposed design also provides superior tracking over P-PI cascade control. In implementing both controllers, neither friction nor lead error compensation was used. Since PPC + LS has higher loop gain, it also has a tendency to amplify the effect of lead errors further. These errors, which can be considered as ‘output’ type disturbances [7], are typically smaller than the errors caused by cutting forces. However, depending on the application of a machine tool, a different compromise may need to be struck between sensitivity to cutting forces versus lead errors and sensor noise. Then, the design procedure outlined in this paper can be applied accordingly. 4. Conclusions This paper has presented a new method for designing feed drive controllers, by directly optimizing the cutting force disturbance response of the electro-mechanical system. The guidelines are straightforward and intuitive. The new control law displays superior rejection of cutting force disturbances compared to the established P-PI cascade architecture. The tracking performance is also very good. Future work targets investigating and augmenting the robustness of this control law. Acknowledgments The authors gratefully acknowledge the financial support of NSERC and CFI, equipment contributions from Heidenhein, NSK, Omron, and New Way Precision, and the assistance of University of Waterloo CNC technician, Mr. Robert Wagner. References
Fig. 9. Table positioning errors during machining.
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