(#14) Crane Operator Performance Comparing PD-Feedback Control andInput Shaping Using Batch-Gradient Descent

I. INTRODUCTION

The efficient use of cranes is a necessity to many manufac-turing, construction, and shipping port activities. Improvingcrane efficiency could have a large impact on a variety ofindustries. For example, payload oscillation increases taskcompletion time, and also increases the likelihood that a craneoperator will run into an obstacle. Thus, reducing payloadoscillation could help ensure safe and efficient operation ofa crane because obstacles would be more easily avoided.

This paper introduces a method to estimate the position of aswinging payload as it is moved through an obstacle course. Adouble–pendulum model can be used to represent the crane asshown in Fig. 1. The frequencies and the associated amplitudesdepend heavily on the payload configuration and cable length[1]–[3]. PD-Feedback Control and input shaping were used tocontrol the payload.

II. RELATED WORK

Many researchers have employed feedback methods tocontrol payload oscillation [4]. The changing human propertiescan make it difficult to tune computerized auxiliary feedbackcontrol systems. Input shaping is a control method that reducespayload oscillation by filtering the human-operator commands[5], [6]. This approach has several advantages over feedbackcontrol. One main advantage is that the human operator isthe sole feedback controller. As such, it is compatible withhuman operators, as supported by the results from numerouscrane-operator studies [3], [7], [8]. A thorough investigationof comparison of PD-Feedback control and input shaping oncranes has been done in [9], [10]. The goal of this paper isto extend the previous work done on these two papers byproviding a quantitative analysis of using the two methods forcrane control through batch gradient descent using trainingdata.

III. APPROACH

Ten operators were asked drive the ten-ton crane to movethe payload from the initial starting point to the final endlocation as shown in Fig. 3. This was to be done in the smallestamount of time with least collisions for each of the threecases: unshaped, feedback and input shaped. The cable lengthwas set to 3.5m and the payload weight 50lbs throughout theexperiment. The position data acquired from a camera placedon the trolley as shown in Fig. 2 was used as a training dataset.

It is desirable for the payload position to be the same asthe trolley position and hence we expect a linear relation-ship between the two. The hypothesis function is given byhθ(x) = θ0x0 + θ1x1 + θ2x2 + θ3x3 where x0 is all ones,x1 is the x co-ordinate of the trolley position, x2 is the y

γ

φ

Trolley

Hook, mh

Payload, mp

l2

l1

u(t)

Fig. 1. The Crane can be modeled as a Double Pendulum

Fig. 2. Ten ton Crane located in the Hi-Bay Area in MARC building

co-ordinate of the trolley position and x3 is the cable length.Let us assume that there are m number of such samples and 3features. The features were scaled so gradient descent wouldconverge quickly. The x and y co-ordinates of the payloadposition are represented by y1 and y2 respectively. This canbe written as Y = θTX.

Hence, we are interested in getting Y given X, where

Y =

[y1

y2

]

X =

x0

x1

x2

x3

θT =

[θ01 θ11 θ21 θ31

θ02 θ12 θ22 θ32

]The gradient descent algorithm used for the multivariate

linear regression model is shown below.Repeat until convergence {

θj ← θj − α∂J(θ)

∂θjfor j = 0, j = 1, j = 2, j = 3

}

Minimizing the cost function J(θ) for m number of suchsamples results in minimizing the error between the trolleyand payload positions.The cost function J , is given by,J(θ) = J(θ0, θ1, θ2, θ3) =

12m

∑mi=1(hθ(x

(i))− y(i))2

Nominal Path

Start End

1.50 m

3.00 m

1.75 m

1.00 m

0.25 m

Obstacles

0.50 m

Fig. 3. The feedback obstacle course used for the evaluation, shown fromthe camera’s point of view as it pointed down at the payload (not shown). Thepayload can be moved in the x and y directions. The task is completed whenthe payload is navigated from the start (green circle) through the nominal pathto the end (red circle).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Hoo

k po

sitio

n (m

)

Trolley position (m)

Training dataLinear regression

Fig. 4. Payload vs Trolley positions for Unshaped case

Code for the entire project was written in MATLAB. Ini-tially, gradient descent was carried out on a one variable linearregression model where only the x co-ordinates of the trolleyand payload positions were taken into consideration. Figure 4shows the training data set for Operator 1 for the trolley andpayload position for the unshaped case. A linear fit providesan estimate of the payload position given any trolley positionfor a constant cable length and weight in the same direction.Sample operators moves are provided in Figs. 5, 6, 7.

The multi-variable linear regression model was able toprovide the x and y co-ordinates of the payload position, giventhe trolley positions and the cable length using the trainingdata.

IV. EVALUATION

The results were evaluated by comparing the payload po-sition and trolley position. The difference between the trolleyposition and the payload position gives the error which needsto be minimal. Given that both PD-feedback and input shapedcontrol techniques reduce oscillation, it was interesting to

End

-2

-1.5

-1

-0.5

0

0.5

1

0 1 2 3 4

TrolleyHook

Posi

tion

(m

)

Position (m)

Start

Fig. 5. Operator 1 moving the payload across Obstacle Course (Unshaped)

-2

-1.5

-1

-0.5

0

0.5

1

0 1 2 3 4

TrolleyHook

Posi

tion

(m

)

Position (m)

StartEnd

Fig. 6. Operator 1 moving the payload across Obstacle Course (Feedback)

-2

-1.5

-1

-0.5

0

0.5

1

0 1 2 3 4

TrolleyHook

Posi

tion

(m

)

Position (m)

StartEnd

Fig. 7. Operator 1 moving the payload across Obstacle Course (Input Shaped)

compare the errors of feedback and shaped. It was speculatedthat input shaping data for this supervised learning algorithmwill provide better estimates than PD-feedback control basedon the data collected in [9], [10].

The payload position was estimated for a given trolleyposition, say x1 = 2m and x2 = 2m with a cable-length,x3 = 3.5m. These distances were chosen because the payloadwould hit the obstacle around this region in Fig. 3. Theestimated payload positions are shown in Fig. 8 for eachoperator. It should be noted that the desired payload positionin this case should have been around 2.828m (=

√8). The

errors in the x and y direction are shown in Fig. 9 and Fig.10 respectively for each operator. The average error in theunshaped case in the x-direction was 1.2m, while feedback andinput-shaped fared better with 0.21m and 0.26m respectively.The average error in the unshaped case in the y-direction was0.71m, while that of feedback and input-shaped was 0.25mand 0.27m respectively.

The table below shows the payload estimates for different

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 Unshaped Payload estimate

Feedback Payload Estimate

Input Shaped Payload Estimate

Fig. 8. Estimated Payload Position

0

0.5

1

1.5

2

2.5 Unshaped Error

Feedback Error

Input Shaped Error

Fig. 9. Error in Estimated Payload Position in X-direction

0

0.5

1

1.5

2

2.5

Unshaped Error

Feedback Error

Input Shaped Error

Fig. 10. Error in Estimated Payload Position in Y-direction

trolley positions for Operator 1 along the obstacle course. Thecable length was all set to 3.5m with constant payload. Thelast row estimates the payload position when the trolley is farout of the obstacle course.

Various Payload Estimates with constant cable lengthMethod Trolley Bridge Payload Payload

Position Position Esimate Estimate(X) (Y) (X) (Y)(m) (m) (m) (m)

Unshaped 2 2 2.528729 1.742516Feedback 2 2 2.043836 1.860681Input Shaped 2 2 1.885045 1.836573Unshaped 3 1.5 3.451711 1.28142Feedback 3 1.5 3.02182 1.377214Input Shaped 3 1.5 2.866315 1.356344Unshaped 4.5 3 4.990178 2.739345Feedback 4.5 3 4.499494 2.828468Input Shaped 4.5 3 4.343648 2.815151Unshaped 10 10 10.733886 9.518119Feedback 10 10 9.924765 9.600703Input Shaped 10 10 9.764155 9.616876

Since there was no data for any other cable length, the modelwas not able to accurately estimate the payload position forany other length. For example, a change in cable length to 5mresulted in poor estimates as shown in the table below becausethere was no data for the gradient descent to converge to.

Method Trolley Bridge Payload PayloadPosition Position Esimate Estimate(X) (Y) (X) (Y)(m) (m) (m) (m)

Unshaped 2 2 21.249582 -1.172001Feedback 2 2 -3.261334 -0.700993Input Shaped 2 2 17.668975 6.535092

V. DISCUSSIONThe purpose of this paper was to take into account all the

features to provide a good estimate of payload positions for ageneralized case. One of the drawbacks of the paper is that itdoes not take the weight of the payload into account. Varyingweights can result in varying payload configurations. Anotherlimitation of the results is that the position of the payload doesnot necessarily have to be exactly same as the trolley positionas long as the payload reaches its final end location in the leastamount of time with the smallest residual vibration. The resultsprovided indicate that feedback does a good job of tracking,but does not take into account the time taken to move thepayload from the initial start point to the end point. Hence,a control technique that only tracks well will not necessarilybe the best approach if it takes a long time to reach its endgoal. The data that was used for this paper involved only tenoperators with the same payload weights and a constant cablelength. Future work would involve adding time and payloadweights as two additional features. While making simulatedmodels for crane control, it is very difficult to take the ”humanfactor” into account. Therefore, a perfect theoretical controlmodel does not necessarily work well when implementedand operated by humans. A larger dataset would presumablyimprove the accuracy of the estimates. The ultimate goalwould be to understand how humans control cranes by usingestimates of payload oscillation to come up with better controlstrategies to mitigate payload swing and improve efficiency.

REFERENCES

[1] R. Blevins, Formulas for Natural Frequency and Mode Shape. NewYork, NY: Van Nostrand Reinhold Co., 1979.

[2] W. Singhose, D. Kim, and M. Kenison, “Input shaping control of double-pendulum bridge crane oscillations,” Journal of Dynamic Systems,Measurement, and Control, vol. 130, no. 3, pp. 1 – 7, May 2008.

[3] D. Kim and W. Singhose, “Performance studies of human operatorsdriving double-pendulum bridge cranes,” Control Engineering Practice,vol. 18, no. 6, pp. 567 – 576, 2010.

[4] E. M. Abdel-Rahman, A. H. Nayfeh, and Z. N. Masoud, “Dynamicsand control of cranes: A review,” JVC/Journal of Vibration and Control,vol. 9, no. 7, pp. 863 – 908, 2003.

[5] O. Smith, Feedback Control Systems. New York: McGraw-Hill BookCo., Inc., 1958.

[6] N. C. Singer and W. P. Seering, “Preshaping command inputs to reducesystem vibration,” Journal of Dynamic Systems, Measurement, andControl, vol. 112, pp. 76–82, March 1990.

[7] A. Khalid, J. Huey, W. Singhose, J. Lawrence, and D. Frakes, “Humanoperator performance testing using an input-shaped bridge crane,” Jour-nal of Dynamic Systems, Measurement and Control, vol. 128, no. 4, pp.835 – 841, 2006.

[8] J. Vaughan and W. Singhose, “Input shapers for reducing overshootin human-operated flexible systems,” in Proceedings of 2009 AmericanControl Conference, St. Louis, MO, June 10-12 2009.

[9] J. Vaughan, A. Karajgikar, and W. Singhose, “A study of crane operatorperformance comparing pd control and input shaping,” in AmericanControl Conference, San Francisco, CA, June 2011.

[10] A. Karajgikar, J. Vaughan, and W. Singhose, “Double-pendulum craneoperator performance comparing pd-feedback control and input shap-ing,” in Advances in Computational Multibody Dynamics, Brussels,Belgium, 4-7 July 2011.