Printer Friendly

Sensorless anti-swing control for automatic gantry crane system: model-based approach.

Abstract

To achieve a good control performance of automatic gantry crane system, sensors are indispensable instrument for feedback signals. However, sensing the payload motion of a real gantry crane, particularly swing motion, is troublesome and often costly. Therefore, sensorless anti-swing controls for automatic gantry crane system are proposed in this paper. The proposed sensorless anti-swing controls are based on mathematical model of the crane based on two approaches. First, a soft sensor based on mathematical model of the crane is introduced to substitute the real swing sensor. In this method, the swing motion of the payload is estimated based on the mathematical model from the measured trolley position. Second, a reference modifier is introduced to produce anti-swing trajectory while performing trolley positioning. An experimental study using lab-scale automatic gantry crane is carried out to evaluate the effectiveness of the proposed sensorless anti-swing controls. The results show that both proposed sensorless anti-swing controls are effective for payload swing suppression since it gives similar performance to the sensor-based anti-swing control. Moreover, the proposed methods are also robust to deal with parameter variations. However, the first method has better performance than the second method.

Keywords: anti-swing, gantry crane, model-based, sensorless.

Introduction

Cranes are widely used in various applications such as heavy loads transportation and hazardous materials handling in shipyards, in factories, in nuclear installations and in high building constructions. Gantry crane is a common type of cranes used to transfer the payload from one position to desired position. A gantry crane incorporates a trolley which moves along the track and translates in a horizontal plane.

In gantry crane system, the load suspended from the trolley by cable is subject to swing caused by improper control input and disturbances. The failure of controlling crane may cause accident and may harm people and the surroundings. Therefore, the gantry crane control must be able to move the trolley adequately fast and to suppress the payload swing at the final position. This is so-called anti-swing control.

Various attempts of anti-swing control for automatic gantry crane have been proposed. Singhose et al., [1] and Park et al., [2] adopted input shaping technique which is open loop approach. However, these methods could not damp the residual swing well. Gupta and Bhowal [3] also presented simplified open-loop anti-swing technique. They have implemented this technique based on velocity control during motion. Other notable researches into time-optimal open-loop control have also been done by Manson [4] and also by Auernig & Troger [5] to control an overhead crane with hoisting. However these are still open-loop approach which is sensitive to system parameters.

On the other hand, feedback controls which are well known to be less sensitive to parameter variations and disturbances have also been proposed in some researches varying from conventional PID (proportional + integral + derivative) to intelligent approaches. Omar [6] proposed PD controls for both trolley position and swing suppression. Nalley & Trabia [7] adopted fuzzy logic control to both positioning control and swing damping. Similarly, Lee & Cho [8] proposed feedback control using fuzzy logic. A fuzzy logic control system with sliding mode control concept was also developed for an overhead crane system by Liu et al. [9]. Furthermore, a fuzzy-based intelligent gantry crane system has also been proposed by Wahyudi & Jalani [10]. The proposed fuzzy logic controllers consist of position as well as anti-swing controllers. The performance of the proposed intelligent gantry crane system had been evaluated experimentally on a lab-scale gantry crane. It was shown that the proposed system has a good positioning performance as well as a good capability to suppress the swing angle in comparison with the crane controlled by the PID controllers.

However, most of the feedback control system proposed needs sensors for measuring the trolley position as well as the load swing motion. In addition, designing the swing measurement of the real gantry crane system, in particular, is not an easy task since there is a hoisting mechanism on parallel flexible cable. Altafini et al. [11] presented a method using measurements of electrical torque and angular velocity of the drives for dynamic load observer. However, it used instead two additional sensors to observe swing angle by knowing the length of the cable.

Some researches have also focused on control schemes with vision system that is more feasible because the vision sensor is not located at the load side. The more recent feedback control using CCD camera was also successfully done by Lee et al. [12] and Osumi et al. [13]. The drawbacks of the vision system, among those are difficult maintenance and high cost [14].

To overcome this problem, sensorless anti-swing control strategies are developed and proposed for automatic gantry crane system. The proposed methods are based on mathematical model of the crane. The real sensor measuring load swing angle is physically omitted in the proposed methods. There are two approaches to achieve the sensorless anti-swing control strategies. In the first method, a soft sensor based on mathematical model of the crane is introduced to substitute the real swing sensor. In this method, the motion of the payload is estimated based on the dynamic model from the measured trolley position. In the second method, a reference modifier is introduced to produce anti-swing trajectory while performing trolley positioning.

In order to evaluate the effectiveness of the proposed methods, experimental study is carried out on a lab-scale gantry crane and their performances are compared with that of sensor-based anti-swing control. The experimental result has shown that the proposed sensorless anti-swing controls are effectively used to suppress the swing motion of the payload since they give similar performance to that of sensor-based anti-swing control well. Furthermore, robustness of the proposed methods to parameter variations is also studied experimentally. It has been shown through experiment that the proposed methods are also robust to parameter variations (i.e. rope length). However, the first method is better than the second method in their performances.

Basic Concept of Sensorless Anti-Swing Control

Most of automatic gantry crane controls proposed by researchers use two controllers as shown in Figure 1, for controlling both trolley position and swing of the crane payload. As no-swing motion of the payload is required, the schema of the feedback system for automatic gantry crane can be simplified as shown in Figure 2. In this feedback control system, two sensors are needed to measure the trolley position X(s) and swing angle [THETA](s). However as discussed previously, swing angle measurement of the real gantry crane system, in particular, is not a simple task.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Therefore, sensorless anti-swing controls are proposed. This can be achieved by two different approaches. First, swing angle is estimated. This is done by soft sensor approach. A soft sensor or virtual sensor is basically a system model designed to substitute the momentary or permanent unavailability of a real sensor in the controlled object [15]. A model-based soft sensor is adopted to estimate the swing motion. Therefore, mathematical model of the crane is needed. This will be called as Sensorless 1 throughout the paper and the schematic diagram is shown in Figure 3.

Second, reference input is modified in order to suppress the payload swing. The modified reference input produces trajectory such that the trolley moves from a point to another point without causing an excessive swing motion of the payload. The mathematical model of the crane system including the characteristic of oscillation is also required to generate the modified reference input. The control diagram is shown in Figure 4. This will be named as Sensorless 2.

[FIGURE 4 OMITTED]

In this paper, the proposed methods are designed based on mathematical model. Mathematical model of gantry crane is derived using Lagrange equations. The model includes also the DC motor dynamic as actuator. Figure 5 shows the simplified diagram of trolley crane mechanism. The position of swinging payload suspended by cable with respect to horizontal and vertical reference, respectively [x.sub.m], and [y.sub.m] can be represented as:

[x.sub.m] = x + lsin[theta] (1)

[y.sub.m] = -lcos[theta] (2)

[FIGURE 5 OMITTED]

The Lagrange equation is developed from derivation of kinetic and potential energy. For simplicity, some assumptions are made. Friction force existing in the trolley is neglected; the cable elongation due to tension force is also neglected. By regarding x and 9 as the generalized coordinates, the Lagrange equations associated to linear and rotational motion are:

([m.sub.1] + [m.sub.2])[??] + [m.sub.1]l([??]cos[theta] - [[??].sup.2] sin[theta]) = F (3)

l[??] + [??] cos[theta] + g sin [theta] = 0 (4)

where F is the summation of all external forces for linear motion and T is the summation of all external torques in rotational motion.

The linearization can be done by considering the swing angle is kept small during control so that sin [theta] [approximately equal to result, (3) and (4) can be written as:

([m.sub.2] + [m.sub.1)[??] + [m.sub.1]l[??] = F (5)

[??] + l[??] + g[theta] =0. (6)

The translational motion of trolley is driven by DC motor. To obtain the entire model of lab-scale crane, the motor dynamic is modelled according to equivalent DC motor circuit. The motor torque, T, is proportional to armature current, i and the back emf voltage proportional to rotational velocity, [omega].

T = [K.sub.t] x i (7)

[e.sub.b] = [K.sub.e][omega] = [K.sub.e.][[??].sub.m]. (8)

The equivalent circuit has armature resistance, R and inductance, L. According to Kirchhoff's law, the relationship between voltages and current can be expressed as follows:

V = R x i + L di/ dt + [e.sub.b]. (9)

The crane system consists of pulleys mechanism to transfer rotational torque of motor shaft to translational trolley motion. With r denotes radius of pulley, J is total moment of inertia and b represents friction constant, the total torques can be summarized.

[K.sub.t] x i - b [[??].sub.m] - Fr = J[[??].sub.m]. (10)

Finally, (5)-(10) can be combined in the form of the following transfer functions:

X(s) / U(s) = [k.sub.0] / s([a.sub.2][s.sup.2] + [a.sub.1]s + 1) (11)

[THETA](s) / X(s) = -[s.sup.2] / l[s.up.2] + g (12)

where:

[k.sub.0] = [K.sub.t]r / [K.sub.e][K.sub.t] + Rb (12.a)

[a.sup.2] = [Lm.sub.2][r.sup.2] + LJ / [K.sub.e][K.sub.t] + Rb (12.b)

[a.sub.1] = [Rm.sub.2][r.sup.2] + RJ + Lb / [K.sub.e][K.sub.t] + Rb. (12.c)

Development of Model-based Sensorless Anti-swing Control

A. Sensorless 1

A model-based soft sensor is proposed to provide output estimation of the payload swing. The schematic diagram is shown in Figure 6, the dynamic information from trolley position X(s) is given to the model-based soft sensor. The model-based soft sensor produces output estimation of the payload swing that will be used for feedback signal to the controller. Consequently, linearized dynamic equation expressed in (6) is used as model-based soft sensor. The swing angle of payload is estimated by using the following:

[??](s) = -[s.sup.2] / [ls.sup.2] + g X(s) (13)

where [THETA](s) and X(s) are estimated swing motion of the payload and trolley motion in Laplace domain respectively.

[FIGURE 6 OMITTED]

B. Sensorless 2

In this method, the controller should produce command input that guarantee the positioning performance while cancelling the payload oscillation especially during acceleration/deceleration. This can be achieved by modifying the input reference. In order to develop modified reference input, according to (6), there is linearized relationship between swing angle and trolley acceleration as follows:

l[??] + g[theta] = -[??]. (14)

Moreover, by differentiating both sides of (1), the following is obtained:

[[??].sub.m] - [??] = l[??]. (15)

Then, (15) is substituted to (14) resulting in the following equation:

[[??].sub.m] + g/l [x.sub.m] - g / l x = 0. (16)

By assuming the feedback control system of Figure 4 has a high bandwidth so that X(s) = [[X.sup.mod.sub.r](s), (16) becomes:

[X.sub.m](s) = g/l / [s.sup.2] + g/l [X.sup.mod.sub.r](s) (17a)

[X.sup.mod.sub.r](s) = [s.sup.2] + g/l / g/l [X.sub.m](s). (17b)

Let's assume there is no input modifier ([X.sub.r] (s) = [X.sup.mod.sub.r] (s)), (17a) can be written as:

[X.sub.m](s) = g/l / [s.sup.2] + g/l [X.sub.r](s). (18)

Equation (18) shows a second order system without damping which gives an oscillation response of the payload position [X.sub.m](s) for any input reference [X.sub.r](s). Theoretically, the oscillatory motion can be suppressed by adding enough damping ratio. To add the damping factor to the system, a reference modifier with modifier parameter K is inserted to the system so that (18) becomes:

[X.sub.m](s) = g/l / [s.sup.2] + Ks + g/l [X.sub.r] (19)

Equation (19) may be written in standard second order system as follows:

[X.sub.m](s)= [[omega].sup.2.sub.n] / [s.sup.2] + 2[zeta][[omega].sub.n]s + [[omega].sup.2.sub.n] [X.sub.r](s) (20)

where:

[[omega].sub.n] = [square root of g/l] (20.a)

K = 2[zeta][[omega].sub.n]. (20.b)

Then, by combining (17a) and (19), the relationship between the original reference input [X.sub.r](s) with the modified reference input [X.sup.mod.sub.r] (s) can be obtained as follows:

[X.sub.r](s)=[X.sup.mod.sub.r](s) + Ks / [s.sup.2] + g/l [X.sup.mod.sub.r](S). (21)

Substituting [X.sup.mod.sub.r](s) in the second term of (21) by (17b), yields:

[X.sub.r](s) = [X.sup.mod.sub.r] (s)+ Kl / g s [X.sub.m](s). (22)

By using (14) and (15), (22) is re-written to the following form:

[X.sub.r](s) = [X.sup.mod.sub.r] (s) - Kl [THETA](s) / s. (23)

Finally, (12) is used to modify (23) becomes:

[X.sup.mod.sub.r] (s) = [X.sub.r](s) - (K s / [s.sup.2] + g/l) X(s). (24)

Figure 7 shows the diagram of the proposed sensorless anti-swing control developed using (24). The proposed modifier parameter K is obtained based on the added damping ratio [zeta] and the natural frequency [[omega].sub.n] as shown in (20).

[FIGURE 7 OMITTED]

Results

A System Description

In order to evaluate the performances of the proposed sensorless anti-swing controls, the proposed methods are implemented to control a lab-scale gantry crane system shown in Figure 8 together with its diagram as shown in Figure 9. The designed lab-scale gantry crane system has four main parts that are trolley system, body frame, potentiometers and a DC motor as an actuator. The DC motor and its driver are used to move the trolley. The DC servo driver circuit operates the motor in the velocity control mode. The input voltage reference between -10.0 volts to 10.0 volts is sent from the PC to drive a 6W, 12V DC motor as control signal for trolley position. To detect trolley position and payload swing angle, 10k[OMEGA] 10-turns and 3/4-turns potentiometers are installed respectively. Noise filters are also included to reduce noisy signals from the potentiometers/sensors. This is done by digital filtering in the PC. The proposed method is implemented digitally on a personal computer and is operated with 1 ms sampling time. The MathWork's MATLAB/Simulink is used for real-time controller implementation through RTW and xPC Target. The experimental setup is shown in Figure 10.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

The proposed sensorless anti-swing controls are based on mathematical model of the crane, a mathematical model of the lab-scale gantry crane was developed. Table 1 lists the known parameter values of the system. Since not all parameters are known to obtain the transfer function model of the system, the unknown parameter of the transfer function was identified using integral step response [16]. Detail of the transfer function identification for gantry crane system was discussed in [17]. The obtained model of the crane is as follows:

X(s) / U(s) = 20.12 / 0.016 [s.sup.2] + 0.234s + 1 (25)

[THETA](s) / X(s) = -[s.sup.2] / 60[s.sup.2] + 981 (26)

B. Controller Design

Well-known classical PID controllers are designed and used to evaluate the effectiveness of the proposed sensorless anti-swing controls. The function of the controller is to control the payload position X(s) so that it moves to the desired position [X.sub.r](s) as fast as possible without excessive swing angle [THETA](s). Due to its simplicity, a PID controller is adopted to control the trolley position, while a PD controller is used for anti-swing controller. The PID controller gains are designed and optimized with simulation model by using Simulink response optimization library block. It is a numerical time domain optimizer developed under MATLAB/Simulink environment. Hence the Simulink response optimization library block assists in time-domain-based control design by setting the desired overshoot, settling time and steady state error.

In order to realize fast motion with small overshoot, the PID controller is optimized by considering the following desired specifications:

* Overshoot [less than or equal to] 2

* Settling time [less than or equal to] 5 s

* Steady state error [less than or equal to] [+ or -] 1%

Moreover, in order to suppress the swing angle quickly, the PD controller is optimized based on the following desired specifications:

* Settling time [less than or equal to] 5 s

* Residual swing [less than or equal to] [+ or -]0.05 rad.

Table 2 lists the obtained PID controller parameters as the result of optimization using Simulink response optimization library block.

Sensorless 2, particularly, requires only PID position control. The same PID gains are used. Moreover, the parameter K has to be designed based on the damping ratio added to the system. Whilst the suitable value of design parameter K can be evaluated to obtain the best performance, with known parameters of model, l = 60cm and g = 981cm/[s.sup.2], thus [[omega].sub.n] = 4.04 rad/s. The selection of K value theoretically corresponds to the damping ratio which affects the settling time of oscillation to diminish. In this paper an additional damping ratio of [zeta]= 0.4 is added to the system. Based on (20b), the parameter K of 3.2 is obtained and used in Sensorless 2.

C. Performance Evaluation

The performances of the proposed sensorless anti-swing control methods are compared with those of sensor-based anti-swing control (Sensor-based). The positioning performances are evaluated in term of overshoot, settling time and error. Whilst swing performances are evaluated based on maximum swing amplitude and its settling time.

Figure 11(a) shows the position responses to a 70 cm step input reference while Table 3 lists the detail positioning performance comparison. Figure 11(a) and Table 3 show that the positioning performance of the both Sensorless 1 and Sensorless 2 are similar to those of Sensor-based. In fact, they use same position sensor to detect the trolley motion. However, the use of the input modifier in Sensorless 2 degrades system accuracy since the error is larger than that of Sensor-based system. Further study has to be done to eliminate the negative effect of the reference input modifier to positioning performance.

Figure 11(b) shows the swing angle responses to a 70 cm step input reference while Table 4 lists the detail anti-swing performance comparison. Figure 11(b) and Table 4 show that the anti-swing performances of Sensorless 1, Sensorless 2 and Sensor-based are also similar each other. Therefore, it can be concluded that the proposed model-based sensorless methods can be used effectively for sensorless swing suppression.

[FIGURE 11 OMITTED]

D. Robustness Evaluation

The proposed methods are designed based on the model assumption of fixed cable length and accurate measurement of the cable length. However, in practice, this assumption works when hoisting mechanism is considered only for lifting and lowering the load at initial and final position respectively. This means hoisting is not performed during gantry motion. In addition, inaccurate measurement of cable length may also exits. Therefore, if one expects that the anti-swing works also for varying cable length, the control strategy must be able to deal with.

In order to evaluate the robustness of the proposed methods, a series of experiment using is carried in which the cable length of the crane is varied [+ or -] 10% of the nominal length. Figures 12-13 show the swing angle motion of the payload. According to Figures 12-13, it is shown that there are no much different swing motion for different cable length for both methods. The performances of the proposed methods do not change significantly due to cable length variation. Hence it can be concluded that the proposed model-based sensorless anti-swing controls are robust to small parameter variation (i.e. cable length). However, it seems that Sensorless 1 is more robust than Sensorless 2. In other word, Sensorless 2 is more sensitive to cable length variation than Sensorless 1.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

Conclusion

In the real application of gantry crane, the use of sensors on the load side is impractical, particularly swing angle sensor. Therefore, sensorless approaches of swing suppression are proposed in this paper. In the first method, a model-based soft sensor is developed to substitute real swing angle sensor. There are no sensors on the payload side. Instead, the swing motion of the payload is estimated based on the dynamic model of the crane and the trolley position. In the second method, a reference input modifier is introduced to eliminate real swing angle sensor for automatic gantry crane so that sensorless anti-swing control is also realized. The swing motion of the crane is suppressed by modifying the reference input to the position control system. Implementation of the proposed method on a lab-scale gantry crane confirmed the effectiveness of the proposed methods. It is also confirmed through experiment that that the proposed methods are robust to parameter variations. However, in general, the first method is better than second one in the sense that the error is smaller and more robust to cable length variation.

Acknowledgment

This research is financially supported by Ministry of Science, Technology and Innovation (MOSTI) Malaysia under eSciencefund Grant 03-01-08-SF0037.

References

[1] Singhose, W.E., Porter, L.J. & Seering, W.P. (1997). Input shaped control of a planar gantry crane with hoisting, Proceedings of the American Control Conference. pp. 97-100.

[2] Park, B.J., Hong, K.S. & Huh, C.D. (2000). Time-efficient input shaping control of container crane systems, Proceedings of IEEE International Conference on Control Application. pp. 80-85.

[3] Gupta, S. and Bhowal, P. (2004). Simplified open loop anti-sway technique. Proceedings of the IEEE India Annual Conference (INDICON), pp.225-228.

[4] Manson, G.A. (1982). Time-optimal control of and overhead crane model, Optimal Control Applications & Methods, Vol. 3, No. 2, 1982, pp. 115-120.

[5] Auernig, J.W. & Troger, H. (1987). Time optimal control of overhead cranes with hoisting of the load, Automatica, Vol. 23, pp. 437-447.

[6] Omar, H.M. (2003). Control of gantry and tower cranes, PhD Dissertation, Virginia Polytechnic Institute and State University. Blacksburg, Virginia.

[7] Nalley, M.J. & Trabia, M.B. (2000). Control of overhead cranes using a fuzzy logic controller. Journal of Intelligent Fuzzy System. Vol.8, pp. 1-18.

[8] Lee, H.H. & Cho, S.K. (2001). A new fuzzy-logic anti-swing control for industrial three-dimensional overhead cranes. Proceedings of IEEE International Conference on Robotics & Automation, pp. 56-61.

[9] Liu, D., Yi, J. and Zhoa, D. (2005). Adaptive sliding mode fuzzy control for two-dimensional overhead crane, Mechatronics, pp. 505-522.

[10] Wahyudi and Jalani, J. (2005). Design and implementation of fuzzy logic controller for an intelligent gantry crane system, Proceedings of The 2nd International Conference on Mechatronics, pp. 345-351.

[11] Altafini, C., Frezza, R. & Galic, J. (2000). Observing the load dynamic of an overhead crane with minimal sensor equipment, Proceedings of the 2000 IEEE International Conference on Robotics & Automation. San Francisco.

[12] Lee, J.J., Nam, G.G., Lee, B.K. & Lee, J.M. (2004). Measurement of 3D spreader position for automatic landing system, Proceedings of The 30th Annual Conference of IEEE Industrial Electronics Society.

[13] Osumi, H., Miura, A. & Eiraku, S. (2005). Positioning of wire suspension system using CCD cameras, Proceedings of IEEE International Conference on Intelligent Robots and Systems (IROS).

[14] Kim, Y.S., Yoshihara, H., Fujioka, N., Kasahara, H., Shim. H. & Sul, S.K. (2003). A new vision- sensorless anti-sway control system for container cranes, Industry Applications Conference. Vol. l, pp.262- 269.

[15] Gonzalez, G.D.; Redard, I.P.; Barrera, R. & Fernandez, M. (1994). Issues in soft-sensor applications in industrial plants, Proceedings of the IEEE International Symposium on Industrial Electronics, pp. 380-385.

[16] Dorsey, J. (2002). Continuous and Discrete Control Systems, MGraw-Hill.

[17] Wahyudi and Jalani J. (2005). Modeling and parameters identification of gantry crane system, Proc. of the International Conference on Recent Advances in Mechanical & Materials Engineering.

Mahmud Iwan Solihin and Wahyudi *

Intelligent Mechatronics System Research Group

Department of Mechatronics Engineering

International Islamic University Malaysia, P.O. Box 10. 50728

Kuala Lumpur, Malaysia. E-mail: iwan.mahmud@yahoo.com

* Correspondence Author E-mail: wahyudi@iiu. edu.my
Table 1: List of parameters.

Parameter Description Value

[m.sub.1] Trolley mass 0.25 kg
[m.sub.2] Payload mass 1 kg
l Cable length 60 cm
g Gravitational acceleration 981 cm/[s.sup.2]
r Radius of pulley 2 cm

Table 2: PID controller parameters.

Gains Controller
 Position control Anti-swing control

Proportional, [K.sub.p] 0.17 13.54
Integral, [K.sub.i] -1.67x[10.sup.-4] -
Derivative, [K.sub.d] 0.07 -0.33

Table 3: Positioning performance comparison.

Performance Controller
 Sensor-based Sensorless 1 Sensorless 2
Overshoot (%) 0 0 0
Settling time (s) 2.7 2.8 4.0
Error (cm) 0.77 0.69 2.60

Table 4: Anti-swing performance comparison.

Performance Controller
 Sensor-based Sensorless 1 Sensorless 2

Amplitude (rad) 0.25 0.22 0.23
Settling time (s) 3.9 4.6 5.6
COPYRIGHT 2007 Research India Publications
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2007 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Solihin, Mahmud Iwan; Wahyudi, Solihin
Publication:International Journal of Applied Engineering Research
Article Type:Report
Geographic Code:9MALA
Date:Jan 1, 2007
Words:4357
Previous Article:An effective asynchronous micropipeline using double edge triggered D Flipflop.
Next Article:Frequency-time MMSE equalizer for BER enhancement of MC-CDMA systems over mulipath fading channel.
Topics:


Related Articles
Vertical lathe designed for speed, precision.

Terms of use | Privacy policy | Copyright © 2020 Farlex, Inc. | Feedback | For webmasters