*Corresponding author: 813 Ferst Dr., Atlanta, GA 30332, USA Copyright © 2015 by ASME Tel.: +1 404 385 0668; fax: +1 404 894 9342. E-mail address: Singhose@gatech.edu

Proceedings of the 8th ASME Dynamic Systems and Control Conference DSCC 2015

October 28-30, 2015, Columbus, Ohio, USA

DSCC2015-9849

COMBINED INPUT SHAPING AND FEEDBACK CONTROL OF A DOUBLE-PENDULUM SYSTEM SUBJECT TO EXTERNAL DISTURBANCES

Robert Mar

Georgia Institute of Technology Atlanta, Georgia, USA

Anurag Goyal Georgia Institute of Technology

Atlanta, Georgia, USA

Vinh Nguyen Georgia Institute of Technology

Atlanta, Georgia, USA

Tianle Yang Georgia Institute of Technology

Atlanta, Georgia, USA

William Singhose* Georgia Institute of Technology

Atlanta, Georgia, USA

ABSTRACT A control system combining input shaping and feedback is

applied to a double-pendulum bridge crane subjected to external disturbances. The external disturbances represent naturally occurring forces, such as gusting winds. The proposed control method achieves fast point-to-point response similar to open-loop input-shaping control. It also minimizes transient deflections and disturbance-induced residual swing using the feedback control. Effects of parameters such as the mass ratio of the double-pendulum, suspension length ratio, and the traveled distance were studied via numerical simulation and hardware experiments. The controller effectively suppresses the disturbances and is robust to modeling uncertainties and task variations.

I. INTRODUCTION Cranes are highly flexible and lightly-damped systems. Manipulation of payloads can be challenging in the presence of swing induced by motion of the crane and externally induced disturbances, such as wind forces. These disturbances can cause large-amplitude swing of the payload leading to potentially hazardous workplace conditions and loss of positioning accuracy. Thus, the crane control system, which usually includes a human operator, must be robust to external disturbances. In addition, crane applications require the payload to move from one point to another with low settling time and minimal transient payload swing in order to achieve high throughput.

Input-shaping methods and feedback control have been extensively developed and used for control of flexible systems. Input-shaping methods have been used to mitigate transient oscillation due to the motion of the crane. Input shaping is a type of Finite Impulse Response (FIR) filtering method. It places zeros near the location of flexible poles of the system, leading to pole-zero cancellation [1][2]. The impulse amplitudes in an input shaper are determined by a set of

constraint equations [3][4]. The command signal convolved with the input-shaper impulses suppresses the unwanted excitation due to intended motion. Numerous variations of this method have been demonstrated successfully to reduce payload sway on single-pendulum configurations [5][6][7], and on double-pendulum configurations [8-15]. Shah and Hong [16] suggested that modified input-shaping methods are also effective in mitigating residual oscillations in damped systems with external forces acting on them as observed in underwater fuel transport system for nuclear power plants.

Much of the previous work on control of cranes was focused on single-pendulum configurations. However, many types of payload structures result in double-pendulum dynamics. Simulation based analyses of advanced input-shaping methods on double-pendulum cranes, and the capability to achieve negligible residual vibration are presented in [9][11]. Masoud and Alhazza [15] presented a frequency-modulation approach to tune the controller frequency such that a single mode input shaper can eliminate residual vibration in double-pendulum systems. Tuan and Lee [17] presented simulation based analyses of non-linear robust controllers using sliding mode control for sway reduction in double-pendulum cranes.

Feedback control methods have also been well studied, and implemented to reject external disturbances [18]. These techniques can successfully reject external disturbances, but do not perform particularly well with motion-induced oscillations of the payload [19].

Input-shaping methods can mitigate motion-induced vibrations, but they cannot reject external disturbances [20]. Therefore, a combination of input shaping and feedback control is necessary for precise positioning of the crane under such circumstances. A combination of feedback control and input shaping has been proposed for single-pendulum crane configurations [6][7][20]. This combines the benefit of both techniques and paves the way for robust controller design. The

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controller can be designed to minimize the settling time and transient oscillations of the payload, and reject external disturbances. However, the previous work on the combined control methodology is limited mostly to single-pendulum configurations. Note that feedback control of double pendulums is much more challenging because it is difficult to accurately measure the payload motion.

In this work, a combined input shaping and feedback controller is proposed for double-pendulum bridge cranes. The system dynamics are described in Section II. The mathematical model and controller implementation in simulation is described in Section III. The experimental setup and results are presented in Section IV. The results from robustness analysis are presented in Section V. Finally, a summary is presented.

NOMENCLATURE m1 = Hook mass (kg) m2 = Payload mass (kg) L1 = Cable length (m) L2 = Rigging length (m) u = Trolley acceleration (ms-2) θ1= Hook angle with vertical (rad) θ2= Payload angle with hook (rad) II. SYSTEM DYNAMICS Figure 1 shows a schematic diagram of a planar double-pendulum bridge crane. The motors provide the actuation force for the trolley, and allow motion along the trolley axis. A cable of length L1 is suspended below the trolley and supports the hook of mass, m1. A rigging cable is attached to the hook, measuring L2, and supports a payload of mass m2. Assuming the lengths of cables to be constant, the governing linearized equations of motion are:

θ1(t) = −

g

L1

⎛⎝⎜

⎞⎠⎟θ

1+

gR

L1

⎛⎝⎜

⎞⎠⎟θ

2−

u(t)L1

θ2(t) =

g

L1

⎛⎝⎜

⎞⎠⎟θ

1−

g

L2

+gR

L2

+gR

L1

⎛⎝⎜

⎞⎠⎟θ

2+

u(t)L1

(1)

where θ1 and θ2 are the angles of the two masses, R is the ratio of the payload mass to the hook mass, u(t) is the acceleration of the trolley, and g is the acceleration due to gravity. The linearized frequencies of the double-pendulum system given by Blevins [21] are:

ω1,2 =g2

(1+ R) 1L1

+ 1L2

⎛

⎝⎜⎞

⎠⎟ β

β = (1+ R)2 1L1

+ 1L2

⎛

⎝⎜⎞

⎠⎟

2

− 4 1+ RL1L2

⎛

⎝⎜⎞

⎠⎟

(2)

The frequencies depend on the ratio of the cable lengths and the mass ratio. These parameters are useful in the design of the input shaper.

For constant total length of the pendulum, the low

frequency component (ω1) remains fairly constant with variation in the rigging length, while the second mode (ω2) has a strong dependence on the parameters. Also, Singhose et al. [9] previously found that the second-mode contribution was increasingly significant for double-pendulum systems with smaller mass ratios and length ratios near unity. This showed that any oscillation control scheme would require more robustness to the second vibration mode, implying use of a robust input shaper.

For the case examined in this paper, both the mass ratio (R) and length ratio (L2/L1) were set at 1. The physical parameters of the system are given in Table 1. Using the mean of the linearized frequencies found previously, an extra insensitive (EI) shaper was found to have sufficient range to suppress both oscillation modes, while also having short shaper duration. The average-mode EI shaper is given by:

Aiti

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥ =

0.27 0.46 0.270 0.62 1.24

⎡

⎣⎢⎢

⎤

⎦⎥⎥

(3)

where, Ai are the amplitudes of the impulses, and ti are the time locations of the impulses. Figure 2 shows the shaper sensitivity to mass ratio error for cable and rigging lengths of 0.75m. The horizontal axis shows mass ratio error, or the fraction of model

Table 1: System parameters Parameter Value

Payload Mass (m2), kg 0.69 Hook Mass (m1), kg 0.69 Cable Length (L1), m 0.75

Rigging Length (L2), m 0.75 Max Trolley Velocity, ms-1 0.30

Max Trolley Acceleration, ms-2 1.0

Figure 1: Schematic of double-pendulum system

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mass ratio to actual mass ratio, and the vertical axis displays the vibration relative to the unshaped motion. The shaper results in less than 10% relative vibration for errors well above ±50%. Thus, the EI shaper is robust to the expected modeling uncertainties. III. CONTROL SYSTEM STRUCTURE

MATLAB & Simulink® were used for model development. The Simulink® block diagram is shown in Figure 3. There are 10 blocks in the diagram. Blocks 1 and 2 model the motor actuator limits and generate the actual motor velocity for a given command velocity. Block 3 provides the trolley acceleration, and Block 4 generates a disturbance with variable magnitude and duration. These are input into Block 5 - pendulum dynamics; the output is payload deflection. Payload deflection is multiplied by a proportional gain to generate a part of the command velocity in Block 6. This feedback loop is responsible for damping the payload oscillation. It is important to note that the deflection feedback controller is only using information about the hook (partial state feedback).

Blocks 7 through 10 form the second feedback loop, and generate the other part of the command velocity. Block 7 integrates the actual velocity to obtain position, and Block 8 generates an error. This position error is input to a PD control in Block 9, which is then EI shaped by Block 10. This feedback loop is responsible for driving the trolley to the destination, with minimal induced vibration.

This dual-loop feedback with shaper-in-the-loop controller has some important implications. First, the deflection feedback loop guarantees tunable transient vibration response and zero residual vibration even under a transient or constant disturbance. This could not be achieved with input shaping alone. Secondly, all trolley movements, as a result of trolley positioning, generate negligible vibration via shaping. This could not be achieved with pure feedback control. Finally, the disturbance can be configured to model many types of wind loads (square wave for gusts, step for sustained wind, sinusoid etc.).

Initial controller gains were chosen by balancing settling time and overshoot, and developing suitable gain envelopes. As described previously, the feedback controller has two parts: the proportional-derivative feedback of the trolley position and the proportional feedback of the hook deflection. Gains for each controller were varied independently and the resulting closed-loop systems were simulated. Figure 4 displays a heat-map of settling times as a function of position proportional gain and deflection proportional gain with constant position derivative gain. Gain combinations that result in a low settling

Figure 2: Shaper robustness

Figure 3: Simulation block diagram

Figure 4: Variation in settling time with controller gains

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time are darkly colored, while gain combinations that settle in 10 seconds or longer are colored white. The dark area represents desirable gain combinations. Position derivative gains were investigated by analyzing a multitude of these heat maps.

The gains were further fine-tuned on the experimental setup. Consequently, the gains for the position feedback were chosen to be 0.5 and 0.0625 for proportional and derivative gains, respectively. For the deflection feedback, a proportional gain of 1.14 was chosen. This design point is shown in Figure 4. This gain combination resulted in low transient vibration, satisfactory settling time, and good stability.

IV. EXPERIMENTAL SETUP AND RESULTS

A double-pendulum structure was fabricated and installed in the Advanced Crane Control Laboratory at the Georgia Institute of Technology. The crane is shown in Figure 5. The double-pendulum consists of a metal shaft with adjustable collars to vary the mass of the payload. The payload shaft is connected to the pendulum hook through a permanent magnet. The rigging length and the length of the suspension of the hook can be adjusted, as shown in the close-up view on the right. An overhead camera is used to track the motion of the hook only. For post-experiment data analysis, an additional camera was placed to the side to simultaneously record the motion of the suspended payload.

Figure 6 compares the simulated unshaped-, simulated combined-, and experimental combined-control of the trolley for a 0.75m move and no disturbance. The unshaped velocity command was generated using feedback control only. It was found by adding the trolley position PD controller output (bypassing the shaper in Figure 3) and the hook deflection proportional controller output. The combined controller command was found similarly, except that the position control output was first convolved with the shaper given in (3).

The graphs in Figure 6 show the shaper and feedback loop working together to position the crane trolley and damp the oscillations. The input shaper minimizes deflection due to the initial acceleration. Next, the feedback controller works to dampen the vibration while also moving to the desired position,

settling at the destination within about 8 seconds. Small payload oscillations near the destination are a result of partial state feedback and no direct control of the payload. Nevertheless, the simulated linear system and experimental results show good agreement. The combined controller significantly improves performance over the unshaped case; it exhibits approximately 60% reduced transient vibration and reduces settling time by 3 seconds.

Figure 7 compares the simulated and experimental responses of a payload disturbance while the system is at equilibrium. In both simulation and experiment, the payload was disturbed with an impulse-like force parallel to the trolley axis, such that the hook would initially deflect approximately 10 degrees. The position of the disturbance is shown by the vertical line. In this case, the controller successfully returns the trolley to the initial position, and simultaneously dampens the oscillations. This combined action is completed in about 8 seconds. Again, the experimental and simulated results agree.

Figure 6: Trolley movement without disturbance

Figure 5: Experimental setup (left) overall view (right) close-up

of hook and payload for double-pendulum crane control

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Figure 8 compares the simulated and experimental responses of the controlled trolley with external disturbance during a 0.75 meter movement. Again, the payload was disturbed in the opposite direction of trolley movement by approximately 10 degrees after 2.5 seconds. The disturbance location is shown as a solid vertical line in the payload deflection plot. Although the controller manages to settle, the experimental response is slower than the previous test cases. This can be attributed to limited actuator acceleration, as evident in the saw-tooth-like velocity command occurring after the disturbance was applied, as shown in Figure 8. The trolley could not respond fast enough to decrease the hook deflection to the same extent as the earlier cases.

V. ROBUSTNESS ANALYSIS

A robustness study was also conducted. Various system parameters such as mass ratio, modeling frequency error, move distance, and disturbance magnitude were varied in both simulation studies and experimental tests.

Figure 9 shows the variation in settling time for different mass ratio values. The controller gains were kept constant at the design conditions for all cases, but the input shaper parameters were updated with changes in the linearized frequencies. As expected, higher mass ratios resulted in longer average frequency, longer shaper durations, and longer settling times. This behavior was observed in both simulations and experiments.

Figure 10 shows the results for same mass ratios, but with a constant input shaper. This study shows the robustness of the proposed combined controller to modeling error. It can be observed that the settling time remains approximately constant. This is a result of the over-damped nature of the trolley position response, which dominates the payload damping for the majority of the transient behavior. In other words, the controller is able to damp payload oscillations before arriving at the destination. Again, this behavior is evident in both simulations and experiments, and confirms the robustness predictions that were shown in Figure 2.

Figure 7: Payload disturbance applied at rest

Disturbance

Figure 8: Payload disturbance applied during trolley movement

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Figure 11 shows the controller robustness to variation in trolley move distance. The settling time remains nearly constant for trolley move distances up to about 0.8 meters and increases with trolley move distance afterwards. For short trolley move distances, the damping of oscillations seems to dominate the settling time. For large trolley move distances, the trolley movement dominates.

Figure 12 shows the robustness to disturbances at equilibrium. As the disturbance magnitude increases, so does the actuator effort, maximum trolley displacement, nonlinearity (pendulum dynamics, actuator saturation, etc.), and settling time. This behavior was found in both simulation and experiment. It should be noted that the settling time remained below 8 seconds in all the cases.

CONCLUSIONS A novel control system methodology involving combined

input shaping and feedback control was presented for control of double-pendulum cranes. The proposed controller effectively rejects external disturbances, while also providing rapid point-to-point motion. The controller is also robust to variation in system parameters such as mass ratio, drive distance, and modeling uncertainties.

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Figure 12: Robustness to disturbance magnitude

Figure 9: Robustness to mass ratio with updating shaper

Figure 10: Robustness to mass ratio without updating shaper

Figure 11: Robustness to trolley move distance

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