Effects of Input Shaping on Manual Control of Flexible and Time-Delayed Systems

Abstract

Objective:

The objective was to study the performance of a manual tracking task with system flexibility and time delays in the input channel and to examine the effects of input shaping the human operator’s commands.

Background:

It has long been known that low-frequency, lightly damped vibration hinders performance of a manually controlled system. Recently, input shaping has been shown to improve the performance of such systems in a compensatory-display tracking task. It is unknown if similar improvements are seen with pursuit-display tasks, or how the improvement changes when time delays are added to the system.

Method:

A total of 18 novice participants performed a pursuit-view tracking experiment with a spring-centered joystick. Controlled elements included an integrator, an integrator with a lightly damped flexible mode, and an input-shaped integrator with a flexible mode. The input to these controlled elements was delayed between 0 and 1 s. Tracking performance was quantified by root mean square tracking error, and subjective difficulty was quantified by ratings on a Cooper–Harper scale.

Results:

Performance was best with the undelayed integrator. Both time delay and flexibility degraded performance. Input shaping improved control of the flexible element, with a diminishing benefit as the time delay increased. Tracking error and subjective rating were significantly related. Some operators used a pulsive control strategy.

Conclusion:

Input shaping can improve the performance of a manually controlled system with flexibility, even when time delays are present.

Application:

This study is useful to designers of human-controlled systems, especially those with problematic flexibility and/or time delays.

Keywords

manual tracking input shaping human operator Cooper–Harper rating

Introduction

There are many situations where a human operator attempts to make the output of a system follow a desired trajectory. For example, Figure 1 shows the task of recording an athlete with a tripod-mounted video camera. The goal of the camera operator 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. This activity is similar to eye tracking, where a human keeps a moving object in the center of his or her vision (Jagacinski, 1977). In these activities, the human is an active part of a feedback control system.

Figure 1.

Video camera tracking task.

Figure 2 gives a simplified block diagram of manual camera tracking. The camera operator is represented by the human block with transfer function $G_{h}$ , and the camera’s rotational dynamics is the camera block with transfer function $G_{c}$ . The athlete’s direction relative to the tripod is the reference signal, $r (t)$ , the camera’s actual direction is the camera output, $y (t)$ , the angle between the actual and desired directions is the error, $e (t)$ . The operator’s force on the handle is the command, $u (t)$ . Other manual control tasks include aiming a tank turret (Kleinman & Perkins, 1974; Tustin, 1947), driving an automobile (Bekey, Burnham, & Seo, 1977; Hess & Modjtahedzadeh, 1990), and piloting an aircraft (McRuer & Jex, 1967).

Figure 2.

Block diagram of camera directional control.

Figure 3 shows a different type of tracking task. A crane is used to pick up a payload and move it to a target location. There are obstacles that must be avoided, so the crane operator generates a desired reference path, $r (t)$ , around these obstacles, and uses an input device to make the payload follow the path. Note that in this example, the reference path is generated by the human, not explicitly imposed by the environment. The crane operator functions as both path planner and trajectory controller.

Figure 3.

Overhead view of crane tracking task.

The difficulty of manual tracking is increased by time delays and flexibility in the system. Figure 4 shows a situation where the human operator’s command, $u (t)$ , drives a crane with an oscillatory output, $y (t)$ . Previous studies have found that the operator’s control behavior becomes much more erratic and nonlinear when the controlled element’s dynamics include a relatively low-frequency flexible mode with a damping ratio below 0.35 (McRuer & Krendel, 1962). This paper studies a crane-like flexible mode with a frequency of $ω_{n} = 1.579$ rad/s, and damping ratio of $ζ = 0.1$ . This low-frequency, lightly damped mode significantly limits the tracking ability of the human–machine system.

Figure 4.

Flexibility in a manual tracking task.

Figure 5 depicts a teleoperated machine. The human operator uses a computer to send commands to the machine, and the machine sends video back to the operator. There is a time delay, $τ_{1}$ , between when the command is issued and when it is executed. There may also be a delay, $τ_{2}$ , between the video transmission and reception. This paper examines only delays in the command channel. Previous research has found that as the time delay increases, tracking error and other performance measures degrade (Adams, 1961; Hess, 1984). Some examples of manually controlled systems that may contain both flexibility and time delays include remotely operated cranes, robotic arms, and unmanned aerial vehicles carrying suspended loads (Mansur et al., 2011). Ricard (1994) gives a list of investigations into the effects of time delay on manual tracking.

Figure 5.

Time delay in a manual tracking task.

To reduce the problem of flexibility, this paper applies a vibration-reducing technique called input shaping (Singer & Seering, 1990; Singhose, 2009; Smith, 1957). This technique improved crane operator performance in Khalid, Huey, Singhose, Lawrence, and Frakes (2006) and Kim and Singhose (2010), and improved manual control of a flexible controlled element with a compensatory display in Potter and Singhose (2013). Important follow-up questions include the following: Can input shaping improve manual tracking with flexible systems using a pursuit display? How do time delays affect the benefit of input shaping for flexible systems? How does quantitative tracking performance relate to subjective ratings?

These questions are addressed in this paper using data from an operator experiment. Tracking performance is assessed quantitatively with root mean square tracking error, and subjectively with ratings on a modified Cooper–Harper scale (Cooper & Harper, 1969). Versions of the scale have been used in numerous studies that require human operators to give numerical ratings of mental workload (Gawron, 2008).

Manual Tracking Task

Previous studies have made extensive use of single-axis manual tracking tasks to investigate the control behavior of human–machine systems (McRuer & Krendel, 1957). The basic setup is shown in Figure 6. 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 signal, and the other object is a cursor that represents the controlled element output. The operator’s goal is to make the cursor closely follow the target.

Figure 6.

Single-axis manual tracking task.

To limit anticipatory control behavior, the target motion must appear random to the human operator. Past studies have shown that an unpredictable target motion can be generated by summing five or more sine waves with arbitrary relative phase. The use of summed sine waves is also advantageous because it facilitates frequency domain analysis. Although real-world tracking tasks rarely consist of summed sine waves, using them is a convenient way to generate an unpredictable signal that tests control behavior over a wide range of frequencies simultaneously.

Figure 7 shows two common manual tracking display types. Figure 7a is a compensatory display, where only the error between target and cursor is displayed. Figure 7b is a pursuit display, where both the target and cursor positions are independently displayed. Pursuit display is used in this study.

Figure 7.

Two display types for manual tracking task.

Figure 8 gives an idealized block diagram of the manual control system with a pursuit display (Allen & McRuer, 1979). The human operator’s control action based on the error signal is called compensatory, and is represented by transfer function $G_{e}$ . With a pursuit display, the operator can generate additional control, $G_{r}$ , based directly on the reference signal. This additional information generally allows a pursuit display to yield better tracking performance than a compensatory display (McRuer, 1980). This paper does not attempt to solve for the internal transfer functions $G_{e}$ and $G_{r}$ , and instead focuses on the input/output relationship of the overall human–machine system. For both display types, tracking performance is often quantified by root mean square error,

R M S E = \sqrt{\frac{1}{T} \int_{0}^{T} {[e (t)]}^{2} d t},

Figure 8.

Block diagram of a manual control system with pursuit display.

where $T$ is the trial duration and $e (t)$ is the instantaneous tracking error at time $t$ . Gerisch, Staude, Wolf, and Bauch (2013) have hypothesized that the tracking error has three main components: (a) error caused by human delay time, (b) error caused by human deficiencies in control processing, and (c) error caused by the human ignoring parts of the signal that he or she considers infeasible to track.

Input Shaping

Input shaping is a command filtering method that limits unwanted oscillation (Singer & Seering, 1990; Singhose, 2009; Singhose & Seering, 2009; Smith, 1957; Vaughan, Jurek, & Singhose, 2011). Figure 9 illustrates a fundamental concept used in input shaping. In the top of Figure 9, 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 9 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.

Figure 9.

Two self-canceling impulses.

The two specially timed impulses can be convolved with an arbitrary function, and the resulting function will maintain the vibration-canceling properties of the original impulses. When used in this way, the series of impulses is called an input shaper. Input shapers may contain more than two impulses. The transfer function of a generic input shaper with $n$ impulses is

G_{i s} (s) = A_{1} e^{- t_{1} s} + A_{2} e^{- t_{2} s} + \dots + A_{n} e^{- t_{n} s},

where $A_{i}$ are the impulse amplitudes, and $t_{i}$ are the time locations of each impulse. The impulse amplitudes and time locations are designed using the estimated natural frequencies ( $ω_{m}$ ) and damping ratios ( $ζ_{m}$ ) of flexible modes to be suppressed. The primary disadvantage of input shaping is that it cannot reduce vibration caused by external disturbances. It also creates a slight increase in rise time by delaying part of the command.

There are various types of input shapers that 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, 1957) is obtained. Its transfer function is

G_{z v} (s) = A_{1} + A_{2} e^{- t_{2} s},

where $A_{1}$ , $A_{2}$ , and $t_{2}$ depend on the natural frequency and damping ratio of the flexible mode. Note that the first impulse time is defined as $t_{1} \equiv 0$ . This can always be done without loss of generality. Only the ZV input shaper is used in this paper. For a description of many other kinds of input shapers, see Vaughan, Yano, and Singhose (2008). The input-shaping technique has proven effective on a wide range of machines (Banerjee, 1993; Drapeau & Wang, 1993; Kim & Singhose, 2010; Magee & Book, 1995; Seth, Rattan, & Brandstetter, 1993; Singhose, Bohlke, & Seering, 1996; Sorensen, Singhose, & Dickerson, 2007; Starr, 1985; Tuttle & Seering, 1997).

For this study, input shaping is added between the command and the controlled element, as shown in Figure 10. The result is an effective controlled element that has much less vibration than the controlled element driven without input shaping.

Figure 10.

Effective controlled element with input shaping applied.

Method

Participants

A total of 18 volunteer human operators (17 males and 1 female, between 20 and 31 years of age) were recruited from the student body of the Georgia Institute of Technology. All participants had experience driving cars, and at least some previous experience using a computer joystick. The experiment took approximately 1 hr, and participants were not paid for their participation.

Procedure

Each participant signed a consent form describing the risks and rewards involved in the experiment. The experimental setup was shown in Figure 6. Participants viewed a pursuit display that occupied around 10° visual angle, and they generated control inputs with a spring-centered joystick (Logitech Attack 3 model) with a maximum angle of 20° from vertical. Around 0.5 Nm of torque was required to displace the joystick from its neutral position, and a maximum torque of 1 Nm was required to hold the joystick at 20°.

Participants performed a series of tracking trials, each lasting 115 s. The first 15-s period allowed the participant to become familiar with the cursor dynamics. Only measurements from the final 100 s of the trial were analyzed. After each trial, the user was asked to consult the rating scale shown in Figure 11 and rate the cursor dynamics on a scale of 1 (best) to 10 (worst). This scale is based on the Cooper–Harper rating scale (Cooper & Harper, 1969) that is used to assess aircraft handling qualities. The participant’s rating indicated the amount of mental effort required to attain good or adequate tracking performance. If adequate performance was not attainable, then the score indicated the amount of mental effort required to maintain control over the cursor.

Figure 11.

Modified Cooper–Harper rating scale for cursor handling qualities.

Table 1 shows transfer functions of the three controlled elements in this experiment: an integrator, an integrator with a flexible mode, and a ZV-input-shaped integrator with a flexible mode. Each transfer function includes a variable time delay, $τ$ . The value of gain K was adjusted to allow the cursor’s maximum velocity to match the maximum velocity of the reference signal using reasonable joystick deflections. For this experiment, the gain was set to .

K = 0.3396 \frac{° visual angle / sec}{° joystick deflection} .

Table 1:

Controlled Element Transfer Functions

Controlled Element	Transfer Function, $G_{c} = \frac{Y (s)}{U (s)}$
Integrator	$e^{- τ s} [\frac{K}{s}]$
Integrator with flexible mode	$e^{- τ s} [\frac{K}{s} (\frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}})]$
ZV-input-shaped integrator with flexible mode	$e^{- τ s} [(A_{1} + A_{2} e^{- t_{2} s}) \frac{K}{s} (\frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}})]$

Note. ZV = Zero Vibration.

A flexible mode with a natural frequency of $ω_{n} = 1.579$ $r a d / s$ ( $0.25 H z$ ) and a damping ratio of $ζ = 0.1$ was chosen. This frequency was slow enough to cause tracking difficulty, but not so slow that extremely long trial durations were required to allow multiple oscillations. It is also a frequency that is encountered in real-world tracking tasks: A crane with a 4-meter payload-suspension cable length would oscillate with approximately this frequency. The ZV input shaper for this natural frequency and damping ratio is

G_{z v} (s) = 0.5783 + 0.4217 e^{- 2.00 s} .

Participants performed trials with each of the controlled elements in Table 1 using time delay values of 0, 0.2, 0.4, 0.6, 0.8, and 1.0 s. The trial order was randomized, and participants were not told which element was being used for each trial. Before data were acquired, participants completed two practice trials to get accustomed to the tracking task and different controlled element dynamics. An extremely easy trial (integrator with no time delay) was used to demonstrate a cursor that should be rated 1 or 2 on the Cooper–Harper scale, and an extremely hard trial (flexible system with a long time delay) demonstrated an undesirable cursor that should be rated very poorly. Table 2 gives the frequency content of the reference signal that drove the motion of the target.

Table 2:

Reference Signal Used for Tracking Tasks

	Frequency, $ω_{i}$	Amplitude, $B_{i}$
Wave Number, $i$	(rad/s)	( $°$ visual angle)
1	9.362	0.120
2	5.718	0.256
3	3.456	0.395
4	2.074	0.536
5	1.194	0.688
6	0.691	0.839
7	0.314	1.057
8	0.189	1.198
9	0.031	16.920

Note. $r (t) = \sum_{i = 1}^{9} B_{i} s i n (ω_{i} t + ϕ_{i})$ . $ϕ_{i}$ are randomly generated at the beginning of each trial.

Assumptions and Limitations

All participants were novice operators, meaning they did not have experience with this kind of tracking task. Novice operators generally exhibit lower performance and more variability in performance than highly trained operators. However, the goal of this study is to determine what tracking situations are inherently easy or difficult for humans in general, so using novice operators was appropriate. There is also value in testing the effectiveness of input shaping with novice operators. Previous results indicated that there is no special training required to use input shaping. Performance improvements were achieved immediately, rather than after extensive training (Kim & Singhose, 2010).

Participants had little practice with the tracking task and Cooper–Harper rating scale before performing the data trials. As a result, their tracking performance and rating behavior likely changed over the course of the experiment. Additional practice would have required increasing the experiment duration or eliminating some of the trial conditions. It was decided that the experimental running time was already relatively long for volunteer participants, and it was desirable to test the complete set of trials. The impact of experimental noise from this learning effect was mitigated by randomizing the trial order and testing a large number of participants.

Results

Tracking Performance

Tracking performance was quantified by root mean square (RMS) tracking error during the 100-s test period. The error as a function of time delay for all three controlled elements is shown in Figure 12. The integrator had the lowest error in all cases. For all time delay values, the unshaped flexible system had the largest tracking error, and the addition of input shaping greatly reduced the error. In fact, input shaping improved the flexible system so much that it nearly matched the performance of the integrator.

Figure 12.

Root mean square (RMS) tracking error averaged across 18 operators. Error bars show one standard deviation above and below the mean.

Strong linear relationships between mean RMS tracking error and time delay were observed for the integrator and input-shaped controlled elements with R² values of .996 and .994, respectively. The unshaped flexible element was not well characterized by a line fit. RMS error for the unshaped flexible element trended upward, and then decreased when the delay was increased from 0.8 to 1 s. This decrease made the relative benefit of input shaping smaller for the longest time delay.

Before running these statistical tests, the tracking error data were log transformed to correct for heteroscedasticity (Gotelli & Ellison, 2004). A two-way analysis of variance (ANOVA) found significant main effects and interaction effects between the two factors of time delay and controlled element type. Post hoc comparisons using Tukey’s honestly significant difference (HSD) test found that within each time delay value, tracking error for the unshaped flexible element was significantly (p < .001) larger than error for the integrator and the input-shaped elements. Error for the input-shaped flexible element was significantly (p < .05) larger than error for the integrator element, except with the 1-s time delay, where they were not significantly different (p = .65). The decrease in RMS error between 0.8 and 1 s for the unshaped flexible element was not significant with a .05 significance level (p = .06).

Cooper–Harper Rating

Figure 13 shows the Cooper–Harper ratings for each controlled element. The ratings showed trends similar to RMS tracking error. For all time delay values, the integrator was significantly (p < .05) better than the other two controlled elements, and the input-shaped flexible element was significantly (p < .05) better than the unshaped flexible element. Ratings generally got worse as the time delay increased.

Figure 13.

Cooper–Harper rating median across 18 operators. Error bars show 95% confidence interval about the median.

Nonparametric statistical analyses were used because Cooper–Harper ratings are ordinal. Medians were compared instead of means, and Mann–Whitney U tests with a Bonferroni correction were used for post hoc pairwise comparisons instead of Tukey’s HSD test (Gotelli & Ellison, 2004). Note that by definition the confidence interval cannot go below 1 or above 10 on the scale.

The mean RMS errors from Figure 12 and median Cooper–Harper ratings from Figure 13 for each time delay value are plotted against each other in Figure 14. There was a log linear relationship between the two variables, given by

C H R = 4.3043 \ln (R M S E) + 2.3219,

Figure 14.

Comparison of quantitative performance to subjective rating. Numbers give the time delay value for each point.

where $C H R$ is Cooper–Harper rating and $R M S E$ is RMS tracking error. The dashed curve in Figure 14 shows this function. The R² value for this curve is .971, and the p value is less than .0001, indicating a significant correlation.

Improvement With Input Shaping

Figure 15 displays the average improvement provided by input shaping for all six time delay values. Note that the dotted lines connecting data points are added to improve readability, not to imply interpolation between points. The performance improvement is normalized by the performance of the unshaped flexible element. More precisely, the graph shows

%improvement = 100 \times (\frac{x_{flexible} - x_{input - shaped}}{x_{flexible}})

Figure 15.

Improvement of adding input shaping to flexible element.

at a given time delay value, where $x$ is either mean RMS tracking error or median Cooper–Harper rating. Figure 15 shows that the ZV input shaper improved tracking performance by between 30% and 60% in both the qualitative and quantitative performance measures. This benefit decreased as the time delay increased.

Control Behavior

Operators used a variety of different strategies to control difficult (sluggish and/or oscillatory) controlled elements. Two strategies are illustrated in Figure 16. Some operators used continuous control, shown in Figure 16a. Other operators used an on–off or pulsive control, such as that shown in Figure 16b. These operators used the continuous input device in a “bang–bang” manner. They quickly moved the joystick to its maximum angle, then returned the joystick to its neutral position. To increase the amount of control input, they either used more pulses or increased the duration of the pulses. Around one third of the operators used such on–off inputs, another third used continuous control, and the final third used strategies that do not fit cleanly into either category. It should be emphasized that operators were not told about any special control strategies before or during the experiment.

Figure 16.

Strategies for controlling difficult elements.

Discussion

Potter and Singhose (2013) presented a manual tracking experiment somewhat similar to the current experiment. That previous study has three important differences: (a) a compensatory display was used instead of a pursuit display, (b) fewer operators were tested, and (c) time delays were not considered. Despite the display differences, the RMS tracking error and subjective rating results were similar to the results in Figures 12 and 13 for the case without time delay (τ = 0). The controlled element without flexibility was best, followed closely by the flexible element with input shaping. The flexible element without input shaping was significantly worse than the other two controlled elements.

In the current study, it was found that the system performance generally declined as the time delay was increased—RMS tracking error and subjective difficulty both increased. This degradation with increasing time delay is consistent with previous investigations (Hess, 1984). Time delays require the human operator to generate phase lead to maintain a given level of tracking performance. Generating phase lead increases mental workload and tends to make humans rate a system more poorly (McRuer & Jex, 1967).

Operators used different strategies to cope with difficult tracking conditions. One method was a pulsive control documented previously in Young and Meiry (1965) and Hess (1979). Young and Meiry proposed that human operators use pulsive control in an attempt to ease the mental computation needed to integrate their own control inputs. It is easier to remember a number of consistently sized pulses than it is to mentally integrate a smooth function.

Findings from related tracking studies can be used to predict how choosing a different oscillatory frequency in this experiment might have impacted the results. Potter and Singhose (2013) found that an oscillatory mode with a natural frequency of $ω_{n}$ = 1.25 $r a d / s$ was much more detrimental to tracking performance than a frequency of $ω_{n}$ = 5 $r a d / s$ . Suppressing the slower, more problematic oscillation with input shaping yielded far greater benefits than suppressing the faster oscillation. Another study added flexible modes with $ω_{n}$ = 7.8 $r a d / s$ and $ω_{n}$ = 16 $r a d / s$ to a proportional controlled element (as opposed to the integrator controlled element in this study) and used a forcing function with four sine waves (McRuer & Krendel, 1957). Although the experimental details differed, the results were similar—the lower frequency mode was more detrimental to tracking performance than was the higher frequency mode. It is therefore expected that a system with an oscillatory natural frequency lower than the $ω_{n}$ = 1.579 $r a d / s$ used in this experiment would have benefitted more from input shaping. Future work could also examine the benefits of input shaping as a function of damping ratio of the oscillatory mode.

Conclusions

A manual tracking experiment was used to investigate human operator performance of one-dimensional tracking tasks with different controlled element dynamics. An integrator was tested as an easily controlled “baseline” case. To test the effectiveness of input shaping for improving control of flexible systems, an integrator with a flexible mode and an input-shaped version of the same controlled element were tested. Time delays up to 1 s were added to the input channel. Results showed that operators were best at tracking with the integrator and rated its dynamics as the most desirable. For all tested time delay values, the input shaper decreased the average tracking error and improved the median rating for the flexible system. Some operators naturally used pulsive-type control with difficult elements, a phenomenon found in previous studies.

Key Points

A total of 18 participants performed a 1-hr manual tracking experiment with a variety of controlled elements, including ones with flexibility and time delays.

Both time delays and flexibility were detrimental to tracking performance.

With difficult controlled elements, some operators used the joystick in a bang-bang, “pulsive” manner (a phenomenon previously noted in the literature).

Input shaping improved quantitative and qualitative control of the flexible element for all time delay values.

Subjective ratings were significantly correlated with log-transformed tracking error.

Footnotes

Acknowledgements

The authors would like to thank the Vertical Lift Consortium for their support of this work under Award No. 2011-B-13-T3.1-A1. Special thanks to Rachel Penczykowski, Heather Humphreys, and Amy Pritchett for their assistance with the statistical analyses. This work was supported by the Vertical Lift Consortium.

James J. Potter earned his BS in general engineering from the University of Illinois at Urbana-Champaign in 2005 and his MS in biomedical engineering from the University of Wisconsin–Madison in 2007. He is a PhD candidate in mechanical engineering at the Georgia Institute of Technology, Atlanta.

William E. Singhose earned his BS in mechanical engineering from the Massachusetts Institute of Technology in 1990 and his MS in mechanical engineering from Stanford University in 1992. He earned his PhD in mechanical engineering from the Massachusetts Institute of Technology in June 1997. He is currently an associate professor at Georgia Tech.

References

Adams

J. L.

(1961, December). An investigation of the effects of the time lag due to long transmission distance upon remote control: Phase I -tracking experiments (NASA technical note). National Aeronautics and Space Administration.

Allen

R. W.

McRuer

(1979). The man/machine control interface—Pursuit control. Automatica, 15, 683–686.

Banerjee

A. K.

(1993). Dynamics and control of the WISP shuttle-antennae system. Journal of the Astronautical Sciences, 41, 73–90.

Bekey

G. A.

Burnham

G. O.

Seo

(1977, August). Control theoretic models of human drivers in car following. Human Factors, 19, 399–413.

Cooper

G. E.

Harper

R. P.

Jr. (1969, April). The use of pilot rating in the evaluation of aircraft handling qualities (Tech. Rep. AGARD 567). Moffett Field, CA: Cornell Aeronautical Laboratory.

Drapeau

Wang

(1993, May). Verification of a closed-loop shaped-input controller for a five-bar-linkage manipulator. In IEEE International Conference on Robotics and Automation (pp. 216–221). Atlanta, GA: IEEE.

Gawron

V. J.

(2008). Human performance, workload, and situational awareness measures handbook (2nd ed.). New York, NY: Taylor & Francis.

Gerisch

Staude

Wolf

Bauch

(2013, May). A three-component model of the control error in manual tracking of continuous random signals. Human Factors, 55, 985–1000. doi:10.1177/0018720813480387

Gotelli

N. J.

Ellison

A. M.

(2004). A primer of ecological statistics. Sunderland, MA: Sinauer Associates.

10.

Hess

R. A.

(1979, May). A rationale for human operator pulsive control behavior. Journal of Guidance and Control, 2, 221–227.

11.

Hess

R. A.

(1984, July). Effects of time delays on systems subject to manual control. AIAA Journal of Guidance, Control, and Dynamics, 7, 416–421.

12.

Hess

R. A.

Modjtahedzadeh

(1990, August). A control theoretic model of driver steering behavior. IEEE Control Systems Magazine, 10(5), 3–8. doi:10.1109/37.60415

13.

Jagacinski

R. J.

(1977, August). A qualitative look at feedback control theory as a style of describing behavior. Human Factors, 19, 331–347.

14.

Khalid

Huey

Singhose

Lawrence

Frakes

(2006, December). Human operator performance testing using an input-shaped bridge crane. ASME Journal of Dynamic Systems, Measurement, and Control, 128, 835–841.

15.

Kim

Singhose

(2010, June). Performance studies of human operators driving double-pendulum bridge cranes. Control Engineering Practice, 18, 567–576.

16.

Kleinman

D. L.

Perkins

T. R.

(1974, August). Modeling human performance in a time-varying anti-aircraft tracking loop. IEEE Transactions on Automatic Control, AC-19, 297–306.

17.

Magee

D. P.

Book

W. J.

(1995, June). Filtering micro-manipulator wrist commands to prevent flexible base motion. In American Control Conference (pp. 924–928). Seattle, WA: American Automatic Control Council.

18.

Mansur

Tischler

Bielefield

Bacon

Cheung

Berrios

Rothman

(2011, May). Full flight envelope inner-loop control law development for the unmanned K-MAX. Paper presented at the American Helicopter Society Forum, Virginia Beach, VA.

19.

McRuer

D. T.

(1980, May). Human dynamics in man-machine systems. Automatica, 16, 237–253.

20.

McRuer

D. T.

Jex

H. R.

(1967, September). A review of quasi-linear pilot models. IEEE Transactions on Human Factors in Electronics, HFE-8, 231–249. doi:10.1109/THFE.1967.234304

21.

McRuer

D. T.

Krendel

E. S.

(1957, October). Dynamic response of human operators (Tech. Rep. No. WADC-TR-56-524). Wright-Patterson Air Force Base, Ohio: Wright Air Development Center.

22.

McRuer

D. T.

Krendel

E. S.

(1962, May). The man-machine system concept. Proceedings of the IRE, 50, 1117–1123.

23.

Potter

J. J.

Singhose

W. E.

(2013, January). Improving manual tracking of systems with oscillatory dynamics. IEEE Transactions on Human-Machine Systems, 43, 46–52.

24.

Ricard

G. L.

(1994, May). Manual control with delays: A bibliography. ACM SIGGRAPH Computer Graphics, 28, 149–154.

25.

Seth

Rattan

Brandstetter

(1993). Vibration control of a coordinate measuring machine. In IEEE Conference on Control Applications (pp. 368–373). Dayton, OH: IEEE.

26.

Singer

N. C.

Seering

W. P.

(1990, March). Preshaping command inputs to reduce system vibration. ASME Journal of Dynamic Systems, Measurement, and Control, 112, 76–82.

27.

Singhose

(2009). Command shaping for flexible systems: A review of the first 50 years. International Journal of Precision Engineering and Manufacturing, 10, 153–168.

28.

Singhose

Bohlke

Seering

(1996). Fuel-efficient pulse command profiles for flexible spacecraft. AIAA Journal of Guidance, Control, and Dynamics, 19, 954–960.

29.

Singhose

Seering

(2009). Command generation for dynamic systems. Raleigh, NC: Lulu Press.

30.

Smith

O. J. M.

(1957, September). Posicast control of damped oscillatory systems. Proceedings of the IRE, 45, 1249–1255.

31.

Sorensen

K. L.

Singhose

Dickerson

(2007, July). A controller enabling precise positioning and sway reduction in bridge and gantry cranes. Control Engineering Practice, 15, 825–837. doi:10.1016/j.conengprac.2006.03.005

32.

Starr

G. P.

(1985, March). Swing-free transport of suspended objects with a path-controlled robot manipulator. ASME Journal of Dynamic Systems, Measurement, and Control, 107, 97–100.

33.

Tustin

(1947, May). The nature of the operator’s response in manual control, and its implications for controller design. Journal of the Institution of Electrical Engineers, 94, 190–206.

34.

Tuttle

Seering

(1997, July). Experimental verification of vibration reduction in flexible spacecraft using input shaping. AIAA Journal of Guidance, Control, and Dynamics, 20, 658–664.

35.

Vaughan

Jurek

Singhose

(2011). Reducing overshoot in human-operated flexible systems. ASME Journal of Dynamic Systems, Measurement, and Control, 133, 011010.

36.

Vaughan

Yano

Singhose

(2008, September). Comparison of robust input shapers. Journal of Sound and Vibration, 315, 797–815. doi:10.1016/j.jsv.2008.02.032

37.

Young

L. R.

Meiry

J. L.

(1965, July). Bang-bang aspects of manual control in high-order systems. IEEE Transactions on Automatic Control, 10, 336–341.