Nonlinear Feedback Control of a Dual-Cable Ropeway System

TECHNICAL PAPER

Nonlinear Feedback Control of a Dual-Cable Ropeway System & Qing Dong and Saroj Biswas

Abstract This paper investigates dynamic control of dual-cable transportation systems, where the haul cable drives a concentrated load on a flexible track cable. The objective of this research is to develop a nonlinear feedback control algorithm applied to this unique cable system, which can be applied to the Navy Underway Replenishment System. First, the nonlinear system model is developed which has the form of a second order nonlinear differential equation coupled with a set of algebraic constraints. Then a linearizing feedback control law is presented that drives a concentrated load smoothly from one point to another. The method is illustrated by simulation results.

Introduction Ropeway transportation systems using a stretched flexible cable have been widely used for many years to transport loads. They usually consist of a single span rope simply anchored at each end as their most basic mechanism form. The first indication of transport using rope comes from rugged Asian countries. In early days, men used a rope system (http://www. mines.edu/library/about_ropeways.html) to transfer themselves with their body suspended by a simple harness, such as a basket, to cross chasms as shown in Figure 1. Early ropeway technology and development was led by the Europeans, particularly in Switzerland and the alpine countries. Wybe Adam, a Dutchman, has been credited as the first successful designer of an operational ropeway system in 1644. Today, ropeway transport systems have found new applications following the introduction of stronger cable, the electric drive, and stability controllers. The ropeway system has a complex dynamic behavior introduced by flexible elements with multiple degrees of freedom. & 2011, American Society of Naval Engineers DOI: 10.1111/j.1559-3584.2010.00238.x

Some of the great challenges to stabilization control of a ropeway system are: & the extent of interactions between haul cable

and track cable, & moving a concentrated load smoothly and

safely on a flexible track cable, & potential longitudinal and transverse oscilla-

tions of the payload. Similar to commercial ropeway transportation, the US Navy extensively uses ropeway systems for Underway Replenishment (UNREP) operation at sea between the ships. Navy UNREP dates back to the earliest years of the Navy. The first significant UNREP operation was with the collier USS Marcellus and the Navy warship USS Massachusetts in 1899. Since then the UNREP operation for the US Navy has become a routine, common occurrence for ships at sea. The US Navy is aggressively pursuing major advances (Otto 2001) in technology, concepts, and designs for new generation equipment that will allow underway resupply with fewer people and less time alongside, as well as with greater safety. 2010 #3 & 21

Dual-Cable Ropeway System

Figure 1: The Historical Ropeway

The UNREP procedures for the US Navy are described in detail in the Naval Warfare Publication ‘‘Underway Replenishment’’ NWP 4-01.4 (Department of the Navy, Office of the Chief of Naval Operations 2001). A typical configuration of UNREP cargo STREAM Rig is shown in Figure 2. Work on the exact nature of the static response of a single cable ropeway system is contained in a book by Routh (1891) where the solution of the symmetrically suspended elastic cable under self-weight was presented. For systems with flexible cables, d’Andrea-Novel et al. (1994) used a hybrid model consisting of a combination of ordinary and partial differential equations to represent the trolley motion and the cable oscillations, and proved exponential stabilization under infinite dimensional settings using simple boundary feedback. Similar strong stability results can also be found in (Conrad and Mifdal 1998), in which a more detailed and accurate model of the trolley-cable system has been used. Starting with a hybrid ordinary differential equation–partial differential equation model, modal analysis of cable motions has been investigated in (Joshi and Rahn 1995); a simple feedback control system has also been

Figure 2:

Cargo

Stream Rig

presented that stabilizes several dominant modes of oscillations. A decoupling controller has been presented in (Beliveau et al. 1993), in which a control yoke is located at the cable support point. This method is similar to that of controlling a cable using a boundary control, and minimizes the effects of disturbances. Lau and Low (1994) investigated the effects of trolley motion trajectories on the load pendulation, and showed that a half-sine type velocity trajectory better replicated the real world manually operated trolley velocity trajectory as compared with a trapezoidal-type trajectory. RoboCrane (Lytle et al. 2002) is an innovative cable-driven manipulator invented by the National Institute of Standards and Technology Intelligent Systems Division. The basic RoboCrane is an inverted Stewart platform parallel link manipulator with cables and winches serving as the links and actuators, respectively. This arrangement provides improved load stability over traditional lift systems and enables 6 degrees of freedom payload control. The technical work presented in this paper focuses on modeling and controller design of a dual-cable ropeway transportation system. The haul cable applies a force to a concentrated load, which moves on a flexible track cable in this dual-cable transportation system. The objective of this research was to develop a nonlinear stabilization control algorithm to transport the load from one point to another. The algorithm is applicable to the Navy UNREP System. Positive cargo position, velocity control, and successful stability control during ropeway operation is essential to increasing system transport throughput and capacity, reducing workload, and improving safety. This paper gives details of the analysis and simulation results for the system model and its dynamic response due to a linearizing feedback control system.

Modeling and Controller Design MODELING This research considers a dual-cable ropeway system, with a haul cable to move the payload 22 & 2010 #3

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from one point to another, and a track cable that provides the path along which the payload moves. The track cable is suspended between two rigid points, P1 and P2, which are not necessarily at the same level with respect to the ground, and the track cable has a sagging profile due to the weight of the payload. The system schematic diagram is shown in Figure 3. This research is developed based on normalized dual-cable dynamics. The trolley and payload are considered as a single point mass. The trolley is driven by two weightless haul cables. The trolley carries the payload as it rides on a support cable, called a track cable. The track cable remains fairly straight under high tension. The track cable has a fixed length, and is assumed to be inextensible. The track cable is supported by two fixed points and pendulation is not a consideration in this research. The vertical components of the acceleration and friction are neglected in contrast to their horizontal components. The following notations are adhered to throughout this paper: b G M D X h1, 2 hL l1, 2 y1, 2 T1, 2 F1, 2

the dynamic friction coefficient the acceleration due to gravity combined mass of the trolley and the payload horizontal distance between P1, and P2 trolley horizontal traveling displacement heights of P1, P2 from the ground level reference the distance from the ground level reference to the bottom of the trolley on the track cable the length of the haul cables between the trolley and P1, P2 angles between the haul cables and the horizontal track cables tension control forces applied on the haul cables

The dynamics of the ropeway system are described using Newton’s law as

Figure 3:

System Geometric Diagram

Based on the assumptions listed above, the following geometric conditions should be satisfied x ¼l1 cos y1 D ¼l1 cos y1 þ l2 cos y2 h1 ¼l1 sin y1 þ hL

ð3Þ

h2 ¼l2 sin y2 þ hL l ¼l2 þ l1 The tension control forces F1 and F2 are nor^1 and u ^2 as malized by u ^1 F1 ¼Mg u ^2 F2 ¼Mg u

ð4Þ

The displacement of the payload in the vertical direction will be small if the track cable is tightly stretched, in which case, y1 and y2 are small. It is assumed that the vertical acceleration and friction terms are small with respect to other terms in equation (2). As the payload rides on the track cable by a pulley, tension applied to the payload should be equal, i.e., T1 5 T2. Substituting equation (4) into equation (2) yields T ¼ Mg

^1 sin y1  u ^2 sin y2 1u sin y1 þ sin y2

ð5Þ

Figure 4:

Payload Force Diagram

M€ x ¼ ðT2 þ F2 Þ cos y2  ðT1 þ F1 Þ cos y1  bMx_ ð1Þ M€y ¼ ðT2 þ F2 Þ sin y2 þ ðT1 þ F1 Þ sin y1  bMy_  Mg

ð2Þ

where F1 and F2 are the drive forces introduced by the haul cables, as shown in Figure 4. NAVAL ENGINEERS JOURNAL

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Dual-Cable Ropeway System

where T 5 T1 5 T2. Substituting equation (5) into the system dynamics equation (1), the system model is simplified to a nonlinear second order differential equation cos y2  cos y1 sin y1 þ sin y2 sinðy1 þ y2 Þ ^ þu sin y1 þ sin y2

€ þ bgx_ ¼ g x

ð6Þ

^2 . ^¼u ^1  u where u For the purpose of control design, the payload displacement is further normalized with respect to a normalizing factor, D, the distance between P1 and P2, the support poles of the track cable. This gives the normalized nonlinear system model as 



^¼ ^ þbg x x

g cos y2  cos y1 D sin y1 þ sin y2 1 sinðy1 þ y2 Þ ^ þ u D sin y1 þ sin y2

g cos y1  cos y2 D sin y1 þ sin y2 1 sinðy1 þ y2 Þ ^ þ u D sin y1 þ sin y2

ð10Þ

the resulting dynamics of this system would be linear  

ð7Þ

^2 þ^ ^2 ¼  bg x v x

ð11Þ



^desired ^1 ^3 ¼ x x x 1



^1 ¼ x ^2 x 

^2 ^2 ¼  bg x x g cos y2  cos y1 1 sinðy1 þ y2 Þ ^; þ þ u D sin y1 þ sin y2 D sin y1 þ sin y2 ð8Þ which can be symbolically expressed as ð9Þ

where the functions f(x, y), h(x, y), and c(x, y) are appropriately defined using equation (8) and equation (3).

CONTROLLER DESIGN Feedback control plays a fundamental role in nonlinear system controller design, as it does in linear control. Feedback linearization is one of the nonlinear controller design approaches that algebraically transforms the original nonlinear system model into a fully or partially linear 24 & 2010 #3

^ v¼

^2 ^1 ¼ x x

The normalized system dynamics are thus represented as

x_ ¼ fðx; yÞ þ hðx; yÞu 0 ¼ cðx; yÞ

equivalent model, so that linear control techniques can be applied. Feedback linearization has been a subject of attractive research and development for many years. In this research, the feedback linearization method is used to design a controller for the ropeway system to cancel the nonlinearities in the original nonlinear system so that the closed-loop dynamics will have a desired linear dynamic model. To do so, assuming that the control law uˆ for the original nonlinear system (8) and the control law ^ v of the corresponding linear model are related by

^desired ^1 being the payload displacement with x x 1 error. An integrator is added in the forward path of the control system to change the system configuration to that of a type one system to reduce the steady state error. The normalized closedloop dynamics are expressed in (12) 2  3 2 32 3 ^1 x ^1 0 1 0 x 6  7 6 76 7 6 7 6 6 ^2 7 6x 7 ¼ 0 bg 0 7 54 x 5 4 ^2 5 4  ^ 1 0 0 x 3 ^3 x ð12Þ 2 3 2 3 0 0 6 7 6 7 desired 7v þ 6 0 7 x þ6 4 1 5^ 4 5 ^1 0 1

The equivalent input ^ v can be designed using standard methods of state feedback control of linear systems. For example, one can take ^ ^1 Þ  kd x ^2 þki x ^3 v ¼ kp ð^ xdesired x 1

ð13Þ

where the feedback gains kp, kd, and ki are computed satisfying appropriate design criteria. The feedback control for the original nonlinear sysNAVAL ENGINEERS JOURNAL

tem is then obtained as ^¼D u

  sin y1 þ sin y2 g cos y2  cos y1 ^ v D sin y1 þ sin y2 sinðy1 þ y2 Þ ð14Þ

Implementation of this control law equation (14) requires the measurements of the trolley position and its velocity, which are easily attained. The controller could compute the state ^3 and some auxiliary variables, y1 and variable x y2, by solving equation (3). The closed-loop system under this control law is represented in the block diagram in Figure 5.

There were two control forces applied in opposite directions on the haul cable. The control force F1 was acting while the trolley traveled from P1 to the equilibrium point of the sagging track cable. The magnitude of control force F1 was small because the trolley traveled downwards on the track cable. The main function of the control force F1 was to prevent the trolley from over speeding due to downhill traveling. The control force F2 was relatively large as it had to overcome gravity to pull the trolley uphill as shown in Figure 7. Figure 5:

ClosedLoop Block Diagram

SIMULATION RESULTS

1 Trolley Position and Velocity from Reference

Based on the predefined desired system performance requirements for the linearized system, which are approximately: natural frequency on ¼ 10 Hz, settling time Ts 5 45 seconds with critical damping without any overshoot response, the desired poles were chosen for achieving such system performance as ldesired ¼ ½0:275; 0:08  0:075i. Then the proportional, derivative, and integral gains for the closed-loop system are estimated using the pole placement method, and further refined through extensive simulation as ([0.028 9.177 0.0248]). A closed-loop system simulation is performed for a step change in reference input.

Figure 6:

Trolley Displacement

0.9 0.8 0.7 0.6 0.5 0.4 Trolley Position Trolley Velocity

0.3 0.2 0.1 0 0

Simulation results have verified the effectiveness of the implemented controllers. Response of the closed-loop system is shown in Figure 6 through Figure 10.

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25 Time

30

35

40

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50

0.45

Figure 7: Control Applied on the Haul Cables

0.4

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0.35 Control Forces

As shown in Figure 6, the trolley starts and ends with a slow speed for achieving a smooth velocity profile without any sudden jerk. It reaches the destination within the required time and with no overshot or oscillation. The closed-loop system is controllable and stable. The significance of this feedback linearization control system is that the control system is implemented on nonlinear system dynamics with geometric constraints.

0.3 0.25 0.2 0.15 0.1 Toward Reference Toward Receiving

0.05 0 0

5

10

15

20

25 30 Time

35

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45

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Trolley position and height followed the track cable profile, because the trolley was traveling on the track cable. The track cable support points P1 and P2 were selected at different levels as shown in Figures 8 and 10. Figure 8 shows the trolley height displacement profile. The track cable tension is related to the position of the Figure 8:

0.35

Trolley

Height Trolley Height from Ground Level

0.34 0.33 0.32 0.31 0.3 0.29 0.28 0.27 0.26 0

Figure 9:

5

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20

25 30 Time

35

40

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3.5

Track Ca-

ble Tension

Track Cable Tension

3

2.5

2

1.5

1

0.5 0

5

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25

30

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50

Time

0.5

Figure 10:

0.45 Trolley Position from Ground Level

Trolley Position and Height Relationship

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

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0.2

0.4 0.6 0.8 Trolley Position from Reference

1

trolley on the track cable. The maximum tension of the track cable occurs when the trolley is at the lowest height of the track cable as shown in Figure 9.

Conclusion This paper presents the nonlinear dynamic model of a dual-cable ropeway transportation system. For control purposes, two haul cables are used to drive a concentrated load on a flexible track cable. The system is represented by a second order nonlinear differential equation with geometric conditions. The control system has the form of feedback linearization, and is used in conjunction with the pole placement method. Simulation results show that the closedloop system with the controller drives the load from one point to another very smoothly. The trolley reaches its destination smoothly within the desired time and with zero steady state error. Future research could investigate the development of stochastic controllers to place a payload at a desired position on a tracking cable when the cable is supported on platforms that oscillate randomly. Full state feedback is typically required in feedback linearization controller design. However, it does not guarantee the controller robustness notwithstanding model parameter uncertainties and/or external disturbances. A trajectory following control method could be considered if the magnitude of oscillations is relatively minor and full state could not be measured. The cables’ mechanical properties need to be accounted for in the dynamic reaction of a moving trolley and the system stability controller. These mechanical properties include the size of the cable, the tension applied on the cables, and maximum of span of the track cable. Another interesting aspect of the ropeway research will be the determination of the cable vibration introduced by wind. In addition to that, the high-speed motion of the trolley on the track cable will enhance the cable vibration behavior. Stability control of a ropeway system is a complex dynamic research subject, however, safe operation of ropeway systems is essential for Navy UNREP and other commercial NAVAL ENGINEERS JOURNAL

applications. Continued research is therefore warranted.

ings of the 1995 American Control Conference, Vol. 4, pp. 2820–2824, 1995.

Acknowledgments

Lau, W.S. and K.H. Low, ‘‘Motion analysis of a suspended mass attached to a crane,’’ Computers and Structures, Vol. 52, No. 1, pp. 169–178, 1994.

Some of the technical details describing the cableway system have been provided by the engineers of Naval Surface Warfare Center (NSWC), including Dr. Samuel Doughty, Dr. Greg Anderson, and Mr. Brian Brady. The authors are grateful for their help and assistance with this study. The first author is also grateful to Dr. John Barkyoumb, Director of Research, NSWC Carderock Division, who provided funding for this effort.

Lytle, A.M., K.S. Saidi, and W.C. Stone, ‘‘Development of a robotic structural steel placement system.’’ Proceeding 19th International Symposium on Automation and Robotics in Construction, Washington, DC, September 23–25, 2002. Otto, C., ‘‘Logistics takes higher priority in Navy planning.’’ Sea Power, May 2001. Routh, E.J., A treatise on analytical statics, Cambridge University Press, Cambridge, UK, 1891.

Author Biographies References Beliveau, Y., D. Dixit Shrikant, and T. Dal, ‘‘Dynamic damping of payload motion for cranes,’’ Journal of Construction Engineering and Management, Vol. 119, No. 3, pp. 631–644, 1993. Conrad, F. and A. Mifdal, ‘‘Strong stability of a model of an overhead crane,’’ Control and Cybernetics, Vol. 27, No. 3, pp. 363–374, 1998. d’Andrea-Novel, B., F. Boustany, F. Conrad, and B.P. Rao, ‘‘Feedback stabilization of a hybrid PDE-ODE system: application to an overhead crane,’’ Mathematics of Control, Signals, and Systems, Vol. 7, pp. 1–22, 1994. Department of the Navy, Office of the Chief of Naval Operations. Underway replenishment NWP 4-01.4 (REV.E), Navy Warfare Publication, March 2001. Joshi, S. and C. Rahn, ‘‘Position control of a flexible cable gantry crane: theory and experiment,’’ Proceed-

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Qing Dong received his M.S. degree in Electrical Engineering from Temple University and the B.S. degree in Electro-Mechanical Engineering from Penn State University. He currently works for Naval Surface Warfare Center in Philadelphia as an Electrical Engineer. E-mail: [email protected]. Dr. Saroj Biswas is a Professor in the Department of Electrical and Computer Engineering, Temple University, Philadelphia. He received the Ph.D. in Electrical Engineering from the University of Ottawa in 1985. His current research interests focus on energy harvesting for autonomous systems, robust, nonlinear control with applications to shipboard cranes, unmanned aircraft and vessels, and intelligent systems. E-mail: [email protected].

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