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Effects of Input Shaping on Manual Tracking with Flexible and Time-Delayed System Dynamics1
Proceedings of the 10-th IFAC Workshop on Time Delay Systems The International Federation of Automatic Control Northeastern University, Boston, USA. June 22-24, 2012
Effects of Input Shaping on Manual Tracking with Flexible and Time-Delayed System Dynamics ? James J. Potter ∗ William E. Singhose ∗ ∗
George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]).
Abstract: Manual tracking tasks are more difficult when the system used for tracking has flexibility and time delays. This paper uses a set of operator experiments to quantitatively examine how system flexibility and time delays degrade manual tracking. Additionally, it investigates the effects of adding input shaping to the system. Performance is quantified by root-mean-squared tracking error and subjective rating of difficulty. Keywords: Manual Control; Input Shaping; Command Shaping; Time Delay; Vibration. 1. INTRODUCTION There are many situations where a human operator attempts to make the output of a system follow a desired state. As an example of manual tracking, Figure 1 shows the task of recording an athlete with a tripod-mounted video camera. The camera operator’s goal is to keep the athlete centered in the camera frame. The actual camera direction is compared to its desired direction (pointed directly at the athlete), and corrective actions are made by applying force to the tripod handle. Figure 2 shows a simplified block diagram of manual camera tracking. The camera operator is represented by the human block with transfer function Gh , and the tripod’s rotational dynamics are the controlled-element 1 block with transfer function Gc . The athlete’s direction relative to the camera is the reference input, r(t), the camera’s actual direction is the controlled-element output, y(t), the angle between the actual and desired directions is the error, e(t), and the operator’s force on the handle is the command, u(t). Steering a car is another good example of a manual control activity that has been well studied (Hess and Modjtahedzadeh, 1990).
Fig. 1. Video camera tracking task
Human
Manual tracking is more difficult when there are time delays and flexibility in the system. As an example system, Figure 3 shows a teleoperated robot. The robot transmits video to the human operator, who uses a computer to send commands back to the robot. There is a time delay, τ1 , between when the command is issued and when it is executed. There may be an additional delay, τ2 , between the video transmission and reception. This paper will only examine delays in the command channel. Previous research has found that as the time delay magnitude is increased, tracking error and other performance measures degrade (Adams, 1961).
Controlled Element
Fig. 2. Block diagram of camera directional control
Communications Satellite
Figure 4 shows a situation where the operator’s command drives a flexible system that oscillates. Previous studies have found that the human operator’s behavior becomes much more erratic and nonlinear when the element dynamics include a rela-
Fig. 3. Manually-controlled system with time delay
? This work was partially supported by the U.S. Vertical Lift Consortium. 1 To be consistent with manual control literature, the component that is under human control in a manual tracking task will often be called a “controlled element” instead of a “machine.”
tively low-frequency flexible mode with a damping ratio below 0.35 (McRuer and Krendel, 1962). Human operators tend to decrease their gain to avoid exciting the flexible mode, and the low gain results in degraded tracking performance (McRuer
978-3-902823-04-5/12/$20.00 © 2012 IFAC
Display and Controller
79
Teleoperated Robot
10.3182/20120622-3-US-4021.00046
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Control Pendant
Crane
Display
Human Operator
Payload Fig. 4. Manually-controlled system with flexibility and Krendel, 1957). This paper will use a flexible mode with a frequency of ωn = 1.0 rad/s, and damping ratio ζ = 0.1. This low-frequency, lightly-damped mode will significantly limit the tracking ability of the human-machine system.
Input Device
To address the problem of flexibility, the use of a vibrationreducing technique called input shaping (Singer and Seering, 1990; Smith, 1958, 1957) is investigated. The technique has proven effective on a wide range of machines including cranes (Blackburn et al., 2010; Sorensen et al., 2007; Starr, 1985), robotic arms (Drapeau and Wang, 1993; Grosser and Singhose, 2000; Magee and Book, 1995), coordinate measuring machines (Jones and Ulsoy, 1999; Singhose et al., 1996c), and satellites (Banerjee, 1993; Singhose et al., 1996a; Tuttle and Seering, 1997; Wie et al., 1993), and has been helpful to human operators (Khalid et al., 2006; Kim and Singhose, 2010).
Fig. 5. Manual tracking experiment
Target
Cursor
(a) Compensatory display
The main questions addressed by this paper are (1) is input shaping able to improve performance of the combined humanmachine system for continuous tracking tasks with flexibility, and (2) does input shaping provide more or less help when time delays are present? These questions will be addressed using data from operator experiments and analytical tools from manual control theory. Performance will be assessed using measures of quantitative tracking performance and subjective ratings of tracking difficulty.
(b) Pursuit display
Section 2 describes a generic manual tracking task. Then, Section 3 gives a basic introduction to input shaping. Section 4 gives details about the operator experiments performed for this investigation, and Section 5 analyzes the experimental results.
Fig. 6. Two display types for manual tracking task
Human Controlled Element
2. MANUAL TRACKING TASK An experiment called a manual tracking task has been used extensively to study the continuous control behavior of humans. The basic setup is shown in Figure 5. A human operator views a display and uses an input device (usually a joystick or force stick) to generate commands. There are two objects on the screen: one is a target that represents the reference input, and the other object is a cursor that is the controlled-element output. The operator’s goal is to make the cursor follow the target as closely as possible, much like the video camera tracking example in the introduction.
Fig. 7. Block diagram of a manual control system between target and cursor is displayed. Figure 6b shows pursuit display (Hess, 1981), where the display “window” is stationary, and both the target and cursor positions are independently displayed.
The target motion must be complicated enough to appear random to the human operator, or else the operator can “cheat” by predicting the future behavior of the target. From past studies, it has been shown that the sum of 5 or more sine waves with arbitrary relative phase is unpredictable to human operators. The use of summed sine waves is advantageous because it facilitates frequency-domain analysis.
Figure 7 shows a block diagram of the manual control system. With a compensatory display, the operator can only see the error between target and cursor. Control action that uses this error signal is called compensatory, and is represented by block Ge . With a pursuit display, the operator can generate an additional control action, Gr , based directly on the reference input. This additional information usually allows a pursuit display to yield better tracking performance than a compensatory display (McRuer, 1980; McRuer and Krendel, 1957).
Figure 6 shows two common manual tracking display types. Figure 6a shows compensatory display, where only the error 80
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
It is difficult to determine the transfer functions Ge and Gr experimentally because a pursuit display does not separate the inputs into these two blocks. This paper will not attempt to solve for these internal transfer functions. It will instead focus on the input/output relationship of the overall human-machine system.
Amplitude
First Second Impulse Impulse 1
During the above discussion, it was assumed that the human controller could be modeled as a linear, time-invariant system. Of course, the human operator is not a linear system. The operator is better characterized as a quasi-linear system that has two components – the describing function accounts for the part of the response that is linearly related to the input, and the remnant accounts for the remaining part of the response that cannot be attributed to the linear model (McRuer, 1980). This paper will focus on describing functions, and will ignore the remnant. Previous studies have found that much of the remnant appears to come from fluctuations in the system’s effective time delay, and it has a fairly-constant power with no major peaks at specific frequencies (Allen and Jex, 1972). Most importantly, for most types of tasks and controlled elements, the remnant is relatively small in magnitude (Hess, 2006)
Time
0
Amplitude
-1 1
Response to Both Impulses Time
0
Fig. 8. Two self-canceling impulses
parameters (Singer and Seering, 1990; Singh and Vadali, 1993; Singhose et al., 2008, 1994, 1996b; Vaughan et al., 2008). For this paper, it will be assumed that the natural frequency and damping ratio of the flexible mode are known exactly, so robustness to modeling errors will not be addressed.
3. INPUT SHAPING Input shaping is a command-filtering method that limits unwanted oscillation (Khalid et al., 2006; Kim and Singhose, 2010; Singer and Seering, 1990; Singhose et al., 2008; Singhose and Seering, 2009; Smith, 1958, 1957; Vaughan et al., 2011). Figure 8 illustrates a fundamental concept used in input shaping. In the top of Figure 8, an impulse is applied to a flexible system, and induces a lightly-damped response. A similar response (shown by the dashed line) results when a second impulse is applied a short time later. The bottom of Figure 8 shows the response that results from both impulses. Because the system is assumed to be linear and time-invariant, the two responses combine linearly, and the vibration is eliminated.
Different input shapers are designed using different combinations of performance requirements. By constraining the impulses to be all positive and the residual vibration to be zero when parameter estimates are perfect, a Zero Vibration (ZV) shaper (Smith, 1958) is obtained. Its transfer function is: Gzv = A1 + A2 e−t2 s ,
(2)
where A1 , A2 , and t2 depend on the natural frequency (ωn ) and damping ratio (ζ) of the flexible mode. Only the ZV shaper will be used in this paper. For a description of many other kinds of shapers, see Vaughan et al. (2008, 2009).
The two specially-timed impulses can be convolved with any arbitrary function, and the resulting function will maintain the vibration-canceling properties of the original impulses. The series of impulses is called an input shaper.
Input shapers can be designed to suppress multiple flexible modes (Hyde and Seering, 1991; Singh and Heppler, 1993; Singhose et al., 2008, 1997). In addition, many studies of crane operators have shown that input shaping can greatly improve performance (Khalid et al., 2006; Kim and Singhose, 2010; Manning et al., 2010). The primary disadvantage of input shaping is that it cannot reduce vibration caused by external disturbances.
The transfer function of a generic input shaper is: Gis = A1 e−t1 s + A2 e−t2 s + · · · + An e−tn s ,
Response to First Impulse Response to Second Impulse
(1)
where Ai are the impulse amplitudes, and ti are the time locations of each impulse. Without loss of generality, the first impulse time is t1 ≡ 0. The impulse amplitudes and time locations are designed using the estimated natural frequencies and damping ratios of flexible modes to be suppressed. Input shapers can be made robust to errors and changes in these
Input shaping is added between the command and the controlled element, as shown in Figure 9. The result is an effective controlled element that has much less vibration than the controlled element alone.
Human Effective Controlled Element Input Shaper
Fig. 9. Effective controlled element with input shaping applied 81
*
Controlled Element
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
4. OPERATOR EXPERIMENT
Table 1. Controlled-element transfer functions
Five human subjects performed the manual tracking experiment described in Section 2. Subjects viewed a pursuit-type display, and generated control inputs with a spring-centered joystick. Each trial lasted 120 seconds. The first 20-second period allowed the operator to get accustomed to the controlled-element dynamics. Only measurements from the final 100 seconds of the trial were analyzed. After each trial, the user was asked to rate the subjective difficulty of the control task by giving a numerical score between 1 (easiest) and 10 (hardest).
Controlled Element
Kc e−τ s s
Integrator
Third-Order System with Flexible Mode
ZV-Input-Shaped Third-Order System with Flexible Mode
Table 1 shows the transfer functions of controlled-element types that were used: an integrator, a third-order system with a flexible mode, and an input-shaped third-order system with a flexible mode. The flexible modes have a natural frequency ωn = 1.0 rad/s and damping ratio ζ = 0.1. Each transfer function includes a variable time delay, τ .
2 ωn 2 s2 +2ζωn s+ωn
(A1 + A2 e−t2 s )
Kc e−τ s s
2 ωn 2 s2 +2ζωn s+ωn
Kc e−τ s s
Table 2. Reference input used for tracking tasks 2 Wave Number, i
Frequency, ωi (rad/s)
Amplitude, Bi (◦ visual angle)
1 2 3 4 5 6 7 8 9 10 11
10.996 8.105 5.781 3.958 2.576 1.571 0.880 0.440 0.189 0.094 0.031
0.181 0.272 0.374 0.487 0.616 0.765 0.939 1.146 1.401 1.609 93.030
Subjects performed 9 trials, using each of the three controlled elements in Table 1 with time delay values of 0, 0.5, and 1.0 second. The trial order was randomized, and the subject was not told which element was used for each trial. The experiment lasted approximately 30 minutes. Before the real trials, subjects completed several practice trials to get accustomed to the task and different controlled-element dynamics. Two of the practice trials showed the subject how hard and easy the control task could be. An extremely easy trial (integrator with no time delay) was defined as a difficulty of 1, and an extremely hard trial (flexible system with longest time delay) was defined as a difficulty of 10. Subjects were not informed what controlled elements were used as the “easy” and “hard” examples.
r(t) =
Table 2 shows the frequency content of the reference input that drove the motion of the target. The range of sine wave frequencies is similar to those from previous manual tracking experiments, except for sine wave 11. This extremely slow wave was included to give the target an overall baseline motion from side to side. From pilot studies, it was found that without this baseline motion, subjects would sometime stop making control inputs to leave the cursor in one place, knowing that the target would eventually come back to it. This strategy was mostly used with the very difficult controlled-elements.
P11 i=1
Bi sin(ωi t + φi )
φi are randomly generated at the beginning of each trial
Integrator Flexible System Input-Shaped Flexible System
Amplitude (dB)
40
Controlled-Element Dynamics Figure 9 shows a Bode plot of the controlled elements with no time delay (τ = 0). To generalize the analysis, the frequency axis has been normalized by the flexible mode’s natural frequency, ωn , and the amplitude curve has been normalized so that the open-loop gain is 1 (0 dB). Note that at ω = ωn , the flexible system curve has a large peak where the phase curve passes through -180◦ . This system tends to be highly oscillatory, unless the human operator is able to improve the response by generating phase lead, reducing gain, or both. In contrast, the corresponding input-shaped amplitude curve has no peak at the oscillatory frequency. The operator can use more aggressive inputs without fear of exciting the oscillatory mode.
20 0 -20 -40 10-1
100
101
Phase (deg)
-90 -180 -270 -360
When a time delay is added to the controlled element, the magnitude is unchanged, but phase lag is added. The lag is small for low frequencies, and increases linearly with frequency. The larger the time delay, the greater the phase loss. 2
Transfer Function
10-1 100 Frequency Ratio (w /w
m)
Fig. 9. Bode diagram of controlled elements
Table format inspired by Table II in Shirley and Young (1968).
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10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Assumptions and Limitations
Integrator Flexible System Input-Shaped Flexible System
For this study, it was assumed that the operators gave full attention to the control task, and there were no major drug or fatigue-related factors that degraded performance. It was further assumed that their joystick command inputs did not exhibit nonlinearities such as saturation, rate-limiting, etc. These are reasonable assumptions based on the recorded time histories of command inputs.
RMS Error (° Visual Angle)
30
An important caveat of this experiment is that all subjects were novice, meaning they did not have previous experience with this tracking task. Novice operators tend to exhibit poor overall tracking performance and relatively large variability in performance between operators, and even between trials. Using more trials, longer trial times, and multiple experimental sessions would have given the subjects more experience with the task, and increased consistency and quantitative performance. However, there is value in demonstrating the effectiveness of input shaping with novice operators. Positive results indicate that there is no special training required to use input shaping, and performance improvements can be achieved immediately, rather than after extensive training.
25 20 15 10 5 0
= 0 sec
= 0.5 sec
= 1 sec
Fig. 10. Root-mean-squared (RMS) tracking error
Integrator Flexible System Input-Shaped Flexible System
5. RESULTS
10 Difficulty Rating
The mean values and standard deviations for each performance metric were computed across all 5 subjects. Error bars on all graphs show one standard deviation above and below the mean. Tracking Performance Results Figure 10 shows the average tracking performance as quantified by root-mean-squared (RMS) error during the 100-second test period. For all three time delay values, the flexible system showed a large RMS error. Input shaping was able to greatly reduce the mean error. For each value of time delay, the mean input-shaped error was around 1/3 of the error that occurred without input shaping. As expected, the pure integrator had the least error.
8 6 4 2 0
= 0 sec
= 0.5 sec
= 1 sec
Fig. 11. Subjective difficulty rating
The tracking error generally got larger as the time delay increased. The integrator and input-shaped elements showed a consistent upward trend, while the tracking error with the unshaped element had a similar error between the 0.5 and 1.0 second delays.
mode, and an input-shaped third-order system with a flexible mode. Time delays of 0, 0.5, and 1 second were added between the operator’s command and the application of the commands to the system. Results showed that the pure integrator always had the lowest error, and was subjectively rated as the easiest. It was also found that for all three time delay values, input shaping greatly decreased root-mean-squared tracking error, and decreased the difficulty rating.
Task Difficulty Results Figure 11 shows the subjective ratings of task difficulty for each system. The difficulty ratings have trends similar to RMS error. The integrator was rated easiest in all cases, and the unshaped flexible element was always most difficult. Two interesting results are that the integrator had a similar difficulty with the 0.5 and 1.0 second delays, and the rating of the unshaped difficulty actually decreased when the time delay was increased from 0.5 to 1.0 second.
REFERENCES Adams, J.L. (1961). An Investigation of the Effects of the Time Lag Due To Long Transmission Distance Upon Remote Control: Phase I - Tracking Experiments. Number D-1211 in NASA Technical Note. National Aeronautics and Space Administration. Allen, R.W. and Jex, H.R. (1972). A simple fourier analysis technique for measuring the dynamic response of manual control systems. IEEE Trans. Syst. Man Cybern., SMC-2(5), 638–643. Banerjee, A.K. (1993). Dynamics and control of the WISP shuttle-antennae system. J. Astronaut. Sci., 41(1), 73–90.
6. CONCLUSIONS Techniques from manual control theory were used to characterize a human performing a one-dimensional tracking task with several different kinds of controlled-element dynamics, including a pure integrator, a third-order system with a flexible 83
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Singhose, W., Bohlke, K., and Seering, W. (1996a). Fuelefficient pulse command profiles for flexible spacecraft. AIAA J. of Guid. Control Dyn., 19(4), 954–960. Singhose, W., Crain, E., and Seering, W. (1997). Convolved and simultaneous two-mode input shapers. IEE Control Theory Appl., 144(6), 515–520. Singhose, W. and Seering, W. (2009). Command Generation for Dynamic Systems. Lulu Press, Inc. Singhose, W., Seering, W., and Singer, N. (1996b). Input shaping for vibration reduction with specified insensitivity to modeling errors. In Japan-USA Sym. Flexible Autom., volume 1. Boston, MA. Singhose, W., Singer, N., and Seering, W. (1996c). Improving repeatability of coordinate measuring machines with shaped command signals. Precis. Eng., 18, 138–146. Smith, O.J.M. (1958). Feedback Control Systems. McGrawHill Book Co., Inc., New York, NY. Smith, O.J.M. (1957). Posicast control of damped oscillatory systems. Proc. IRE, 45, 1249–1255. Sorensen, K.L., Singhose, W., and Dickerson, S. (2007). A controller enabling precise positioning and sway reduction in bridge and gantry cranes. Control Eng. Pract., 15(7), 825– 837. Special Issue on Award Winning Applications, 2005 IFAC World Congress. Starr, G.P. (1985). Swing-free transport of suspended objects with a path-controlled robot manipulator. J. Dyn. Syst. Meas. Control, 107(1), 97–100. Tuttle, T. and Seering, W. (1997). Experimental verification of vibration reduction in flexible spacecraft using input shaping. AIAA J. of Guid. Control Dyn., 20(4), 658–664. Vaughan, J., Jurek, P., and Singhose, W. (2011). Reducing overshoot in human-operated flexible systems. J. Dyn. Syst. Meas. Control, 133(1), 011010. Vaughan, J., Yano, A., and Singhose, W. (2008). Comparison of robust input shapers. J. Sound Vib., 315(4-5), 797–815. Vaughan, J., Yano, A., and Singhose, W. (2009). Robust negative input shapers for vibration suppression. J. Dyn. Syst. Meas. Control, 131(3), 031014. Wie, B., Sinha, R., Sunkel, J., and Cox, K. (1993). Robust fuel- and time-optimal control of uncertain flexible space structures. In AIAA Guid. Navig. Control Conf., 939–948. Monterey, CA.
Blackburn, D., Singhose, W., Kitchen, J., Patrangenaru, V., Lawrence, J., Kamoi, T., and Taura, A. (2010). Command shaping for nonlinear crane dynamics. J. Vib. Control, 16(4), 477–501. Drapeau, V. and Wang, D. (1993). Verification of a closed-loop shaped-input controller for a five-bar-linkage manipulator. In IEEE Int. Conf. Rob. Autom., 216–221. Atlanta, GA. Grosser, K. and Singhose, W. (2000). Command generation for reducing perceived lag in flexible telerobotic arms. JSME Int J., Ser. C, 43(3), 755–761. Hess, R.A. and Modjtahedzadeh, A. (1990). A control theoretic model of driver steering behavior. IEEE Control Syst. Mag., 10(5), 3–8. Hess, R.A. (1981). Pursuit tracking and higher levels of skill development in the human pilot. IEEE Trans. Syst. Man Cybern., SMC-11(4), 262–273. Hess, R.A. (2006). Feedback Control Models – Manual Control and Tracking, chapter 38, 1249–1294. John Wiley & Sons, Inc., Hoboken, NJ, 3 edition. Hyde, J. and Seering, W. (1991). Using input command preshaping to suppress multiple mode vibration. In IEEE Int. Conf. Rob. Autom., volume 3, 2604–2609. Sacramento, CA. Jones, S. and Ulsoy, A.G. (1999). An approach to control input shaping with application to coordinate measuring machines. J. Dyn. Syst. Meas. Control, 121(2), 242–247. Khalid, A., Huey, J., Singhose, W., Lawrence, J., and Frakes, D. (2006). Human operator performance testing using an inputshaped bridge crane. J. Dyn. Syst. Meas. Control, 128(4), 835–841. Kim, D. and Singhose, W. (2010). Performance studies of human operators driving double-pendulum bridge cranes. Control Eng. Pract., 18(6), 567–576. Magee, D.P. and Book, W.J. (1995). Filtering micromanipulator wrist commands to prevent flexible base motion. In American Control Conf., 924–928. Seattle, WA. Manning, R., Clement, J., Kim, D., and Singhose, W. (2010). Dynamics and control of bridge cranes transporting distributed-mass payloads. J. Dyn. Syst. Meas. Control, 132(1), 014505. McRuer, D.T. (1980). Human dynamics in man-machine systems. Automatica, 16(3), 237–253. McRuer, D.T. and Krendel, E.S. (1957). Dynamic response of human operators. Technical Report WADC-TR-56-524. McRuer, D.T. and Krendel, E.S. (1962). The man-machine system concept. Proc. IRE, 50(5), 1117–1123. Shirley, R. and Young, L. (1968). Motion cues in man-vehicle control effects of roll-motion cues on human operator’s behavior in compensatory systems with disturbance inputs. IEEE Trans. Man-Mach. Syst., 9(4), 121–128. Singer, N.C. and Seering, W.P. (1990). Preshaping command inputs to reduce system vibration. J. Dyn. Syst. Meas. Control, 112, 76–82. Singh, T. and Heppler, G.R. (1993). Shaped input control of a system with multiple modes. J. Dyn. Syst. Meas. Control, 115(3), 341–347. Singh, T. and Vadali, S.R. (1993). Robust time-delay control. J. Dyn. Syst. Meas. Control, 115(2A), 303–306. Singhose, W., Kim, D., and Kenison, M. (2008). Input shaping control of double-pendulum bridge crane oscillations. J. Dyn. Syst. Meas. Control, 130(3), 034504. Singhose, W., Seering, W., and Singer, N. (1994). Residual vibration reduction using vector diagrams to generate shaped inputs. J. Mech. Des., 116, 654–659.
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Effects of Input Shaping on Manual Tracking with Flexible and Time-Delayed System Dynamics ? James J. Potter ∗ William E. Singhose ∗ ∗
George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]).
Abstract: Manual tracking tasks are more difficult when the system used for tracking has flexibility and time delays. This paper uses a set of operator experiments to quantitatively examine how system flexibility and time delays degrade manual tracking. Additionally, it investigates the effects of adding input shaping to the system. Performance is quantified by root-mean-squared tracking error and subjective rating of difficulty. Keywords: Manual Control; Input Shaping; Command Shaping; Time Delay; Vibration. 1. INTRODUCTION There are many situations where a human operator attempts to make the output of a system follow a desired state. As an example of manual tracking, Figure 1 shows the task of recording an athlete with a tripod-mounted video camera. The camera operator’s goal is to keep the athlete centered in the camera frame. The actual camera direction is compared to its desired direction (pointed directly at the athlete), and corrective actions are made by applying force to the tripod handle. Figure 2 shows a simplified block diagram of manual camera tracking. The camera operator is represented by the human block with transfer function Gh , and the tripod’s rotational dynamics are the controlled-element 1 block with transfer function Gc . The athlete’s direction relative to the camera is the reference input, r(t), the camera’s actual direction is the controlled-element output, y(t), the angle between the actual and desired directions is the error, e(t), and the operator’s force on the handle is the command, u(t). Steering a car is another good example of a manual control activity that has been well studied (Hess and Modjtahedzadeh, 1990).
Fig. 1. Video camera tracking task
Human
Manual tracking is more difficult when there are time delays and flexibility in the system. As an example system, Figure 3 shows a teleoperated robot. The robot transmits video to the human operator, who uses a computer to send commands back to the robot. There is a time delay, τ1 , between when the command is issued and when it is executed. There may be an additional delay, τ2 , between the video transmission and reception. This paper will only examine delays in the command channel. Previous research has found that as the time delay magnitude is increased, tracking error and other performance measures degrade (Adams, 1961).
Controlled Element
Fig. 2. Block diagram of camera directional control
Communications Satellite
Figure 4 shows a situation where the operator’s command drives a flexible system that oscillates. Previous studies have found that the human operator’s behavior becomes much more erratic and nonlinear when the element dynamics include a rela-
Fig. 3. Manually-controlled system with time delay
? This work was partially supported by the U.S. Vertical Lift Consortium. 1 To be consistent with manual control literature, the component that is under human control in a manual tracking task will often be called a “controlled element” instead of a “machine.”
tively low-frequency flexible mode with a damping ratio below 0.35 (McRuer and Krendel, 1962). Human operators tend to decrease their gain to avoid exciting the flexible mode, and the low gain results in degraded tracking performance (McRuer
978-3-902823-04-5/12/$20.00 © 2012 IFAC
Display and Controller
79
Teleoperated Robot
10.3182/20120622-3-US-4021.00046
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Control Pendant
Crane
Display
Human Operator
Payload Fig. 4. Manually-controlled system with flexibility and Krendel, 1957). This paper will use a flexible mode with a frequency of ωn = 1.0 rad/s, and damping ratio ζ = 0.1. This low-frequency, lightly-damped mode will significantly limit the tracking ability of the human-machine system.
Input Device
To address the problem of flexibility, the use of a vibrationreducing technique called input shaping (Singer and Seering, 1990; Smith, 1958, 1957) is investigated. The technique has proven effective on a wide range of machines including cranes (Blackburn et al., 2010; Sorensen et al., 2007; Starr, 1985), robotic arms (Drapeau and Wang, 1993; Grosser and Singhose, 2000; Magee and Book, 1995), coordinate measuring machines (Jones and Ulsoy, 1999; Singhose et al., 1996c), and satellites (Banerjee, 1993; Singhose et al., 1996a; Tuttle and Seering, 1997; Wie et al., 1993), and has been helpful to human operators (Khalid et al., 2006; Kim and Singhose, 2010).
Fig. 5. Manual tracking experiment
Target
Cursor
(a) Compensatory display
The main questions addressed by this paper are (1) is input shaping able to improve performance of the combined humanmachine system for continuous tracking tasks with flexibility, and (2) does input shaping provide more or less help when time delays are present? These questions will be addressed using data from operator experiments and analytical tools from manual control theory. Performance will be assessed using measures of quantitative tracking performance and subjective ratings of tracking difficulty.
(b) Pursuit display
Section 2 describes a generic manual tracking task. Then, Section 3 gives a basic introduction to input shaping. Section 4 gives details about the operator experiments performed for this investigation, and Section 5 analyzes the experimental results.
Fig. 6. Two display types for manual tracking task
Human Controlled Element
2. MANUAL TRACKING TASK An experiment called a manual tracking task has been used extensively to study the continuous control behavior of humans. The basic setup is shown in Figure 5. A human operator views a display and uses an input device (usually a joystick or force stick) to generate commands. There are two objects on the screen: one is a target that represents the reference input, and the other object is a cursor that is the controlled-element output. The operator’s goal is to make the cursor follow the target as closely as possible, much like the video camera tracking example in the introduction.
Fig. 7. Block diagram of a manual control system between target and cursor is displayed. Figure 6b shows pursuit display (Hess, 1981), where the display “window” is stationary, and both the target and cursor positions are independently displayed.
The target motion must be complicated enough to appear random to the human operator, or else the operator can “cheat” by predicting the future behavior of the target. From past studies, it has been shown that the sum of 5 or more sine waves with arbitrary relative phase is unpredictable to human operators. The use of summed sine waves is advantageous because it facilitates frequency-domain analysis.
Figure 7 shows a block diagram of the manual control system. With a compensatory display, the operator can only see the error between target and cursor. Control action that uses this error signal is called compensatory, and is represented by block Ge . With a pursuit display, the operator can generate an additional control action, Gr , based directly on the reference input. This additional information usually allows a pursuit display to yield better tracking performance than a compensatory display (McRuer, 1980; McRuer and Krendel, 1957).
Figure 6 shows two common manual tracking display types. Figure 6a shows compensatory display, where only the error 80
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
It is difficult to determine the transfer functions Ge and Gr experimentally because a pursuit display does not separate the inputs into these two blocks. This paper will not attempt to solve for these internal transfer functions. It will instead focus on the input/output relationship of the overall human-machine system.
Amplitude
First Second Impulse Impulse 1
During the above discussion, it was assumed that the human controller could be modeled as a linear, time-invariant system. Of course, the human operator is not a linear system. The operator is better characterized as a quasi-linear system that has two components – the describing function accounts for the part of the response that is linearly related to the input, and the remnant accounts for the remaining part of the response that cannot be attributed to the linear model (McRuer, 1980). This paper will focus on describing functions, and will ignore the remnant. Previous studies have found that much of the remnant appears to come from fluctuations in the system’s effective time delay, and it has a fairly-constant power with no major peaks at specific frequencies (Allen and Jex, 1972). Most importantly, for most types of tasks and controlled elements, the remnant is relatively small in magnitude (Hess, 2006)
Time
0
Amplitude
-1 1
Response to Both Impulses Time
0
Fig. 8. Two self-canceling impulses
parameters (Singer and Seering, 1990; Singh and Vadali, 1993; Singhose et al., 2008, 1994, 1996b; Vaughan et al., 2008). For this paper, it will be assumed that the natural frequency and damping ratio of the flexible mode are known exactly, so robustness to modeling errors will not be addressed.
3. INPUT SHAPING Input shaping is a command-filtering method that limits unwanted oscillation (Khalid et al., 2006; Kim and Singhose, 2010; Singer and Seering, 1990; Singhose et al., 2008; Singhose and Seering, 2009; Smith, 1958, 1957; Vaughan et al., 2011). Figure 8 illustrates a fundamental concept used in input shaping. In the top of Figure 8, an impulse is applied to a flexible system, and induces a lightly-damped response. A similar response (shown by the dashed line) results when a second impulse is applied a short time later. The bottom of Figure 8 shows the response that results from both impulses. Because the system is assumed to be linear and time-invariant, the two responses combine linearly, and the vibration is eliminated.
Different input shapers are designed using different combinations of performance requirements. By constraining the impulses to be all positive and the residual vibration to be zero when parameter estimates are perfect, a Zero Vibration (ZV) shaper (Smith, 1958) is obtained. Its transfer function is: Gzv = A1 + A2 e−t2 s ,
(2)
where A1 , A2 , and t2 depend on the natural frequency (ωn ) and damping ratio (ζ) of the flexible mode. Only the ZV shaper will be used in this paper. For a description of many other kinds of shapers, see Vaughan et al. (2008, 2009).
The two specially-timed impulses can be convolved with any arbitrary function, and the resulting function will maintain the vibration-canceling properties of the original impulses. The series of impulses is called an input shaper.
Input shapers can be designed to suppress multiple flexible modes (Hyde and Seering, 1991; Singh and Heppler, 1993; Singhose et al., 2008, 1997). In addition, many studies of crane operators have shown that input shaping can greatly improve performance (Khalid et al., 2006; Kim and Singhose, 2010; Manning et al., 2010). The primary disadvantage of input shaping is that it cannot reduce vibration caused by external disturbances.
The transfer function of a generic input shaper is: Gis = A1 e−t1 s + A2 e−t2 s + · · · + An e−tn s ,
Response to First Impulse Response to Second Impulse
(1)
where Ai are the impulse amplitudes, and ti are the time locations of each impulse. Without loss of generality, the first impulse time is t1 ≡ 0. The impulse amplitudes and time locations are designed using the estimated natural frequencies and damping ratios of flexible modes to be suppressed. Input shapers can be made robust to errors and changes in these
Input shaping is added between the command and the controlled element, as shown in Figure 9. The result is an effective controlled element that has much less vibration than the controlled element alone.
Human Effective Controlled Element Input Shaper
Fig. 9. Effective controlled element with input shaping applied 81
*
Controlled Element
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
4. OPERATOR EXPERIMENT
Table 1. Controlled-element transfer functions
Five human subjects performed the manual tracking experiment described in Section 2. Subjects viewed a pursuit-type display, and generated control inputs with a spring-centered joystick. Each trial lasted 120 seconds. The first 20-second period allowed the operator to get accustomed to the controlled-element dynamics. Only measurements from the final 100 seconds of the trial were analyzed. After each trial, the user was asked to rate the subjective difficulty of the control task by giving a numerical score between 1 (easiest) and 10 (hardest).
Controlled Element
Kc e−τ s s
Integrator
Third-Order System with Flexible Mode
ZV-Input-Shaped Third-Order System with Flexible Mode
Table 1 shows the transfer functions of controlled-element types that were used: an integrator, a third-order system with a flexible mode, and an input-shaped third-order system with a flexible mode. The flexible modes have a natural frequency ωn = 1.0 rad/s and damping ratio ζ = 0.1. Each transfer function includes a variable time delay, τ .
2 ωn 2 s2 +2ζωn s+ωn
(A1 + A2 e−t2 s )
Kc e−τ s s
2 ωn 2 s2 +2ζωn s+ωn
Kc e−τ s s
Table 2. Reference input used for tracking tasks 2 Wave Number, i
Frequency, ωi (rad/s)
Amplitude, Bi (◦ visual angle)
1 2 3 4 5 6 7 8 9 10 11
10.996 8.105 5.781 3.958 2.576 1.571 0.880 0.440 0.189 0.094 0.031
0.181 0.272 0.374 0.487 0.616 0.765 0.939 1.146 1.401 1.609 93.030
Subjects performed 9 trials, using each of the three controlled elements in Table 1 with time delay values of 0, 0.5, and 1.0 second. The trial order was randomized, and the subject was not told which element was used for each trial. The experiment lasted approximately 30 minutes. Before the real trials, subjects completed several practice trials to get accustomed to the task and different controlled-element dynamics. Two of the practice trials showed the subject how hard and easy the control task could be. An extremely easy trial (integrator with no time delay) was defined as a difficulty of 1, and an extremely hard trial (flexible system with longest time delay) was defined as a difficulty of 10. Subjects were not informed what controlled elements were used as the “easy” and “hard” examples.
r(t) =
Table 2 shows the frequency content of the reference input that drove the motion of the target. The range of sine wave frequencies is similar to those from previous manual tracking experiments, except for sine wave 11. This extremely slow wave was included to give the target an overall baseline motion from side to side. From pilot studies, it was found that without this baseline motion, subjects would sometime stop making control inputs to leave the cursor in one place, knowing that the target would eventually come back to it. This strategy was mostly used with the very difficult controlled-elements.
P11 i=1
Bi sin(ωi t + φi )
φi are randomly generated at the beginning of each trial
Integrator Flexible System Input-Shaped Flexible System
Amplitude (dB)
40
Controlled-Element Dynamics Figure 9 shows a Bode plot of the controlled elements with no time delay (τ = 0). To generalize the analysis, the frequency axis has been normalized by the flexible mode’s natural frequency, ωn , and the amplitude curve has been normalized so that the open-loop gain is 1 (0 dB). Note that at ω = ωn , the flexible system curve has a large peak where the phase curve passes through -180◦ . This system tends to be highly oscillatory, unless the human operator is able to improve the response by generating phase lead, reducing gain, or both. In contrast, the corresponding input-shaped amplitude curve has no peak at the oscillatory frequency. The operator can use more aggressive inputs without fear of exciting the oscillatory mode.
20 0 -20 -40 10-1
100
101
Phase (deg)
-90 -180 -270 -360
When a time delay is added to the controlled element, the magnitude is unchanged, but phase lag is added. The lag is small for low frequencies, and increases linearly with frequency. The larger the time delay, the greater the phase loss. 2
Transfer Function
10-1 100 Frequency Ratio (w /w
m)
Fig. 9. Bode diagram of controlled elements
Table format inspired by Table II in Shirley and Young (1968).
82
101
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Assumptions and Limitations
Integrator Flexible System Input-Shaped Flexible System
For this study, it was assumed that the operators gave full attention to the control task, and there were no major drug or fatigue-related factors that degraded performance. It was further assumed that their joystick command inputs did not exhibit nonlinearities such as saturation, rate-limiting, etc. These are reasonable assumptions based on the recorded time histories of command inputs.
RMS Error (° Visual Angle)
30
An important caveat of this experiment is that all subjects were novice, meaning they did not have previous experience with this tracking task. Novice operators tend to exhibit poor overall tracking performance and relatively large variability in performance between operators, and even between trials. Using more trials, longer trial times, and multiple experimental sessions would have given the subjects more experience with the task, and increased consistency and quantitative performance. However, there is value in demonstrating the effectiveness of input shaping with novice operators. Positive results indicate that there is no special training required to use input shaping, and performance improvements can be achieved immediately, rather than after extensive training.
25 20 15 10 5 0
= 0 sec
= 0.5 sec
= 1 sec
Fig. 10. Root-mean-squared (RMS) tracking error
Integrator Flexible System Input-Shaped Flexible System
5. RESULTS
10 Difficulty Rating
The mean values and standard deviations for each performance metric were computed across all 5 subjects. Error bars on all graphs show one standard deviation above and below the mean. Tracking Performance Results Figure 10 shows the average tracking performance as quantified by root-mean-squared (RMS) error during the 100-second test period. For all three time delay values, the flexible system showed a large RMS error. Input shaping was able to greatly reduce the mean error. For each value of time delay, the mean input-shaped error was around 1/3 of the error that occurred without input shaping. As expected, the pure integrator had the least error.
8 6 4 2 0
= 0 sec
= 0.5 sec
= 1 sec
Fig. 11. Subjective difficulty rating
The tracking error generally got larger as the time delay increased. The integrator and input-shaped elements showed a consistent upward trend, while the tracking error with the unshaped element had a similar error between the 0.5 and 1.0 second delays.
mode, and an input-shaped third-order system with a flexible mode. Time delays of 0, 0.5, and 1 second were added between the operator’s command and the application of the commands to the system. Results showed that the pure integrator always had the lowest error, and was subjectively rated as the easiest. It was also found that for all three time delay values, input shaping greatly decreased root-mean-squared tracking error, and decreased the difficulty rating.
Task Difficulty Results Figure 11 shows the subjective ratings of task difficulty for each system. The difficulty ratings have trends similar to RMS error. The integrator was rated easiest in all cases, and the unshaped flexible element was always most difficult. Two interesting results are that the integrator had a similar difficulty with the 0.5 and 1.0 second delays, and the rating of the unshaped difficulty actually decreased when the time delay was increased from 0.5 to 1.0 second.
REFERENCES Adams, J.L. (1961). An Investigation of the Effects of the Time Lag Due To Long Transmission Distance Upon Remote Control: Phase I - Tracking Experiments. Number D-1211 in NASA Technical Note. National Aeronautics and Space Administration. Allen, R.W. and Jex, H.R. (1972). A simple fourier analysis technique for measuring the dynamic response of manual control systems. IEEE Trans. Syst. Man Cybern., SMC-2(5), 638–643. Banerjee, A.K. (1993). Dynamics and control of the WISP shuttle-antennae system. J. Astronaut. Sci., 41(1), 73–90.
6. CONCLUSIONS Techniques from manual control theory were used to characterize a human performing a one-dimensional tracking task with several different kinds of controlled-element dynamics, including a pure integrator, a third-order system with a flexible 83
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Singhose, W., Bohlke, K., and Seering, W. (1996a). Fuelefficient pulse command profiles for flexible spacecraft. AIAA J. of Guid. Control Dyn., 19(4), 954–960. Singhose, W., Crain, E., and Seering, W. (1997). Convolved and simultaneous two-mode input shapers. IEE Control Theory Appl., 144(6), 515–520. Singhose, W. and Seering, W. (2009). Command Generation for Dynamic Systems. Lulu Press, Inc. Singhose, W., Seering, W., and Singer, N. (1996b). Input shaping for vibration reduction with specified insensitivity to modeling errors. In Japan-USA Sym. Flexible Autom., volume 1. Boston, MA. Singhose, W., Singer, N., and Seering, W. (1996c). Improving repeatability of coordinate measuring machines with shaped command signals. Precis. Eng., 18, 138–146. Smith, O.J.M. (1958). Feedback Control Systems. McGrawHill Book Co., Inc., New York, NY. Smith, O.J.M. (1957). Posicast control of damped oscillatory systems. Proc. IRE, 45, 1249–1255. Sorensen, K.L., Singhose, W., and Dickerson, S. (2007). A controller enabling precise positioning and sway reduction in bridge and gantry cranes. Control Eng. Pract., 15(7), 825– 837. Special Issue on Award Winning Applications, 2005 IFAC World Congress. Starr, G.P. (1985). Swing-free transport of suspended objects with a path-controlled robot manipulator. J. Dyn. Syst. Meas. Control, 107(1), 97–100. Tuttle, T. and Seering, W. (1997). Experimental verification of vibration reduction in flexible spacecraft using input shaping. AIAA J. of Guid. Control Dyn., 20(4), 658–664. Vaughan, J., Jurek, P., and Singhose, W. (2011). Reducing overshoot in human-operated flexible systems. J. Dyn. Syst. Meas. Control, 133(1), 011010. Vaughan, J., Yano, A., and Singhose, W. (2008). Comparison of robust input shapers. J. Sound Vib., 315(4-5), 797–815. Vaughan, J., Yano, A., and Singhose, W. (2009). Robust negative input shapers for vibration suppression. J. Dyn. Syst. Meas. Control, 131(3), 031014. Wie, B., Sinha, R., Sunkel, J., and Cox, K. (1993). Robust fuel- and time-optimal control of uncertain flexible space structures. In AIAA Guid. Navig. Control Conf., 939–948. Monterey, CA.
Blackburn, D., Singhose, W., Kitchen, J., Patrangenaru, V., Lawrence, J., Kamoi, T., and Taura, A. (2010). Command shaping for nonlinear crane dynamics. J. Vib. Control, 16(4), 477–501. Drapeau, V. and Wang, D. (1993). Verification of a closed-loop shaped-input controller for a five-bar-linkage manipulator. In IEEE Int. Conf. Rob. Autom., 216–221. Atlanta, GA. Grosser, K. and Singhose, W. (2000). Command generation for reducing perceived lag in flexible telerobotic arms. JSME Int J., Ser. C, 43(3), 755–761. Hess, R.A. and Modjtahedzadeh, A. (1990). A control theoretic model of driver steering behavior. IEEE Control Syst. Mag., 10(5), 3–8. Hess, R.A. (1981). Pursuit tracking and higher levels of skill development in the human pilot. IEEE Trans. Syst. Man Cybern., SMC-11(4), 262–273. Hess, R.A. (2006). Feedback Control Models – Manual Control and Tracking, chapter 38, 1249–1294. John Wiley & Sons, Inc., Hoboken, NJ, 3 edition. Hyde, J. and Seering, W. (1991). Using input command preshaping to suppress multiple mode vibration. In IEEE Int. Conf. Rob. Autom., volume 3, 2604–2609. Sacramento, CA. Jones, S. and Ulsoy, A.G. (1999). An approach to control input shaping with application to coordinate measuring machines. J. Dyn. Syst. Meas. Control, 121(2), 242–247. Khalid, A., Huey, J., Singhose, W., Lawrence, J., and Frakes, D. (2006). Human operator performance testing using an inputshaped bridge crane. J. Dyn. Syst. Meas. Control, 128(4), 835–841. Kim, D. and Singhose, W. (2010). Performance studies of human operators driving double-pendulum bridge cranes. Control Eng. Pract., 18(6), 567–576. Magee, D.P. and Book, W.J. (1995). Filtering micromanipulator wrist commands to prevent flexible base motion. In American Control Conf., 924–928. Seattle, WA. Manning, R., Clement, J., Kim, D., and Singhose, W. (2010). Dynamics and control of bridge cranes transporting distributed-mass payloads. J. Dyn. Syst. Meas. Control, 132(1), 014505. McRuer, D.T. (1980). Human dynamics in man-machine systems. Automatica, 16(3), 237–253. McRuer, D.T. and Krendel, E.S. (1957). Dynamic response of human operators. Technical Report WADC-TR-56-524. McRuer, D.T. and Krendel, E.S. (1962). The man-machine system concept. Proc. IRE, 50(5), 1117–1123. Shirley, R. and Young, L. (1968). Motion cues in man-vehicle control effects of roll-motion cues on human operator’s behavior in compensatory systems with disturbance inputs. IEEE Trans. Man-Mach. Syst., 9(4), 121–128. Singer, N.C. and Seering, W.P. (1990). Preshaping command inputs to reduce system vibration. J. Dyn. Syst. Meas. Control, 112, 76–82. Singh, T. and Heppler, G.R. (1993). Shaped input control of a system with multiple modes. J. Dyn. Syst. Meas. Control, 115(3), 341–347. Singh, T. and Vadali, S.R. (1993). Robust time-delay control. J. Dyn. Syst. Meas. Control, 115(2A), 303–306. Singhose, W., Kim, D., and Kenison, M. (2008). Input shaping control of double-pendulum bridge crane oscillations. J. Dyn. Syst. Meas. Control, 130(3), 034504. Singhose, W., Seering, W., and Singer, N. (1994). Residual vibration reduction using vector diagrams to generate shaped inputs. J. Mech. Des., 116, 654–659.
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