Command Shaping for Flexible Systems: A Review of the First 50 Years William Singhose1,# 1 Woodruff School of Mechanical Engineering, Georgia Institut...

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DOI 10.1007/s12541-009-0084-2

Command Shaping for Flexible Systems: A Review of the First 50 Years William Singhose1,# 1 Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA, 30332 # Corresponding Author / E-mail: [email protected], TEL: +1-404-385-0668 KEYWORDS: Command Shaping, Input Shaping, Vibration, Oscillation

The control of flexible systems is a large and important field of study. Unwanted transient deflection and residual vibration are detrimental to many systems ranging from nano-positioning devices to large industrial cranes. Thousands of researchers have worked diligently for decades to provide solutions to the challenging problems posed by flexible dynamic systems. The work can roughly be broken into three categories:1) Hardware design, 2) Feedback control, and 3) Command shaping. This paper provides a review of command-shaping research since it was first proposed in the late 1950’s. The important milestones of the research advancements, as well as application examples, are used to illustrate the developments in this important research field. Manuscript received: August 12, 2009 / Accepted: September 10, 2009

1. Introduction Flexible dynamic systems suffer from unwanted transient deflection and residual vibration. These detrimental effects cause significant problems for positioning accuracy, throughput, fatigue, and safety for many types of systems ranging from nanopositioning devices to large industrial cranes. Thousands of researchers have diligently worked for decades to provide solutions to the challenging problems posed by flexible dynamic systems. The work can roughly be broken into three categories:1) Hardware design, 2) Feedback control, and 3) Command shaping. This paper provides a review of command-shaping research that has proven useful since it was first proposed in the late 1950’s. The important milestones of the research advancements, as well as application examples, are used to illustrate the developments in this important research field. In order to convey the impact of command shaping on the performance of a flexible system, let us consider an illustrative example – a crane. Cranes are used to perform important and challenging manipulation tasks such as construction of bridges, dams, and high-rise towers. Tower cranes, like the ones shown in Figure 1, are commonly used in construction to provide a large workspace. Cranes used indoors often have the structure of the bridge crane shown in Figure 2. © KSPE and Springer 2009

Fig. 1 Tower Cranes at la Sagrada Família in Barcelona While the physical structure of cranes varies widely, one essential element is constant – an overhead support cable is used to lift and transport the payload. This essential element provides the fundamental usefulness of cranes. However, it also creates one of the biggest problems: payload oscillation. The 10-ton industrial bridge crane shown in Figure 2 is equipped with an overhead vision system that can track the motion

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Unfortunately, at that time, digital computers were a rarity; so implementing command-shaping methods was very challenging (with digital computers, they are very easy to implement – often easier than feedback control). Starting in the 1980’s, microprocessors became widespread and implementation of command shaping became straightforward. However, that advancement was not the only requirement for successful and widespread use of command shaping. Another missing element was robustness to modeling errors, uncertainties, and nonlinearities. All command-shaping methods require some information about the system dynamics. This information will always have some degree of inaccuracy. So, for command shaping to be successful in real applications, it must have an adequate level of robustness. Researchers in the late 1980’s provided a significant breakthrough that created command-shaping methods with good robustness properties. These advancements are reviewed in Section 3. Methods for applying command shaping to multi-mode systems are reviewed in Section 4. Section 5 shows how command shaping is related to time-optimal control. Example applications of command shaping are given in Section 6. Concluding remarks are then given at the end of the paper.

Trolley

Hook Pendent Payload Fig. 2 10-Ton Bridge Crane at Georgia Tech Obstacles Typical Response

Goal

2. Early Command-Shaping Methods

Input-Shaped Response Collisions

Start

Fig. 3 Typical Hook Response of the hook. Figure 3 shows the response of the crane hook for a typical maneuver under standard operation. The human operator was attempting to drive the crane from the Start location to the Goal location, while avoiding the obstacles in the workspace. The oscillation of the crane made it very difficult to move through the obstacle field. As a result, the crane hook collided with two of the obstacles. The oscillations and collisions inhibited safe and efficient operation. Figure 3 also shows the response when the operator enabled command shaping and attempted the task again. In the case with command shaping, the operator was able to drive cleanly through the obstacle field, without oscillation, and complete the task much faster. The operator also experienced less stress and exerted less effort because the crane dynamics were greatly simplified.1-3 The command shaping intercepts the operator commands to the crane and modifies them in real time. The results in Figure 3 are typical of what can be expected with command shaping control – the dynamic response of the flexible system is greatly simplified without the use of added sensors and feedback control, or the need to redesign the mechanical hardware. Section 2 gives a brief overview of the early developments in command shaping. These methods first appeared in the 1950’s.

Perhaps the earliest work on systematic command shaping was performed by OJM Smith in the late 1950’s.4-6 (Elements of command shaping ideas appear even earlier in works directed at cam profile design,7-10 signal component control,11,12 and command smoothing.13-17 However, OJM Smith was the first to provide a systematic description that could be easily followed). Smith’s method, known as posicast control, took a baseline command and delayed part of the command before giving it to the system. The delayed portion of the command canceled out the vibration induced by the portion of the baseline command that was not delayed. As a first step to understanding how this approach can move flexible systems without vibration, it is helpful to start with the simplest command – an impulse. Applying an impulse, A1, to a flexible system will cause it to vibrate. The response of an underdamped system to such an impulse is shown at the top of Figure 4. If a second impulse, A2, is applied at a later time, as shown in the middle of Figure 4, then the vibration induced by the first impulse may be cancelled. This concept is demonstrated at the bottom of Figure 4. Impulse A1 induces the vibration indicated by the dashed line, while A2 induces the dotted response. Combining the two responses results in zero residual vibration. The second impulse must be applied at the correct time and must have the appropriate magnitude for complete cancellation. The posicast method proposed by OJM Smith effectively took two impulses whose vibrations were self-canceling and convolved them with the baseline reference command. This command-shaping process, now usually called input shaping, is demonstrated in Figure 5 using an S-curve as the initial command and two impulses

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f(t)

A1

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Fig. 4 Vibration from Two Impulses Can Cancel

of the original function, one of which is shifted in time. The third, and final, step adds the two functions together to obtain the convolution product. The solid line in Figure 5 shows the convolution product that results from summing the two replicas of the original function. The input-shaping process can be represented in several ways. In addition to the convolution process shown in Figure 5, input shaping can be accomplished by time-delay blocks. Such a block diagram representation of the input-shaping process is shown in Figure 6. The unshaped reference command, f(t), is fed into n gain blocks that correspond to the impulse amplitudes, Ai. The scaled functions are then sent through time delay blocks, ∆i, that correspond to the impulse time locations. Note that the first shaper impulse, A1, is located at time zero, so it does not have an associated time delay block. The shaped command signal, c(t), is formed by summing the scaled and time-delayed functions. The amplitudes and time locations of the impulses in an input shaper can be determined by solving a set of constraint equations. Most types of constraints can be categorized as residual vibration constraints, robustness constraints, impulse amplitude constraints, and the requirement of time optimality. To constrain the residual vibration, we need an expression for the residual vibration amplitude as a function of an impulse sequence. If we assume the system can be modeled as a secondorder harmonic oscillator, then the system response from a single impulse is:

Aω −ζω − y 0 (t ) = 0 e ( 0 ) sin ω 1 − ζ 2 ( t − t0 ) , 2 1 − ζ t

(

t

)

(1)

where A0 is the amplitude of the impulse, t0 is the time the impulse is applied, ω is the natural frequency, and ζ is the damping ratio. The response from a sequence of impulses is the superposition of the response given in (1). Using the simplification: ωd = ω 1 − ζ 2 ,

(2)

the response to a sequence of impulses after the last impulse is: Fig. 5 The Input-Shaping Process

n Aω −ζω t − t yΣ (t ) = ∑ i e ( ) sin (ω d ( t − ti ) ). 2 i =1 1 − ζ i

in the input shaper. Convolving a two-impulse sequence with a continuous function is very easy. First, scale the initial command by the amplitude of the first impulse, A1. Next, form a secondary function by scaling the original command by the amplitude of the second impulse, A2, and shift it to the time location of the second impulse, as shown in Figure 5. There are now two, scaled replicas

(3)

Given (3), an expression for the amplitude of residual vibration can be formed by using the trigonometric identity: n

∑= B sin (ω t + φ ) = AΣ sin (ω t + ψ ) , i

i

1

i

(4)

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1 A 1 + K = t 0

where, 2

2

i

n

i

i

i

(5)

i

Given the expression in (3), the coefficients in the summation of (4) are: B =

i

i

1− ζ 2

e

−ζω ( t − t ) i

.

(6)

To calculate the residual vibration amplitude, we evaluate (5) at the time of the last impulse, t = tn. Substituting (6) into (5) and bringing the constant portion of the coefficients out of the square root term gives: AΣ =

ω

2

2

2

e −ζω n C (ω , ζ ) + S (ω , ζ ) , t

1−ζ

(7)

where, n

C (ω , ζ ) = ∑ Ai eζω t cos (ω d ti ) i

(8)

i =1 n

ζω t S (ω , ζ ) = ∑ Ae sin (ω d ti ). i i

(9)

i =1

The vibration amplitude can be expressed as a non-dimensional function by dividing (7) by the amplitude of residual vibration from a single impulse of unity magnitude. The resulting percentage residual vibration expression gives us the ratio of vibration with input shaping to that without input shaping. By expressing the constraint in this way, we can set the residual vibration to a desired percentage of the vibration that occurs without input shaping. The amplitude of residual vibration from a single unitymagnitude impulse applied at time zero is: A↑ =

ω 1−ζ 2

.

(10)

Dividing (7) by (10) yields the percentage vibration equation:

V (ω , ζ ) =

i

i

Aω

AΣ = e −ζω n [C (ω , ζ )]2 + [ S (ω , ζ )]2 . A↑ t

(11)

If V(ω,ζ) is set equal to zero at the modeling parameters, (ωm, ζm), then a sequence of impulses that satisfies the equation is called a Zero Vibration (ZV) shaper. As with any filtering method, a constraint must be applied to ensure that the shaped command produces the same rigid-body motion as the unshaped command. To satisfy this requirement, the impulse amplitudes must sum to one: n

∑= A = 1. i

i

(12)

1

Due to the transcendental nature of (11), there will be multiple possible solutions. To make the solution time optimal, the time of the final impulse must be minimized:

min ( tn ) ,

, 2 ω 1−ζ

i

AΣ = ∑ B cos (φ ) + ∑ B sin (φ ) . =1 =1 n

K 1+ K π

(13)

When the above constraints are solved for a two-impulse sequence, the ZV shaper is obtained as:5,18

(14)

where, K =e

−ζπ

1−ζ 2

.

(15)

The duration of the ZV shaper (the time location of its final impulse) is equal to one half period of the damped vibration. One of the initial challenges of implementing ZV shaping was creating the correctly scaled and time-delayed component of the shaped command. Given that analog computers were used during this time period, the time delay was particularly challenging to obtain. Therefore, many of the papers on ZV shaping that followed OJM Smith’s work concentrated on implementation issues.19-22 In practice, ZV shapers can be sensitive to modeling errors. To demonstrate this effect, the amplitude of residual vibration can be plotted as a function of modeling errors. Figure 7 shows such a sensitivity curve for the ZV shaper. The vertical axis is the percentage vibration given by (11), while the horizontal axis is the normalized frequency formed by dividing the actual frequency of the system, ωa, by the modeling frequency, ωm. Notice that the residual vibration increases rapidly as the actual frequency deviates from the modeling frequency. The robustness can be measured quantitatively by measuring the width of the curve at some low level of vibration. This non-dimensional robustness measure is called the input shaper’s frequency insensitivity, Iω. The 5% insensitivity of the ZV shaper is 0.06, as shown in Figure 7. This means that the shaper can keep the residual vibration below the 5% level for frequency changes of only ±3 percent. Given the sensitivity to modeling errors, parameter uncertainties, and nonlinearities that is demonstrated by the ZV sensitivity curve, the usefulness of the early command-shaping methods was limited to applications where the frequencies were well known and did not change significantly during operation. In order to make command shaping widely applicable, this weakness needed to be overcome. Luckily, many such methods have been developed since the late 1980’s.

3. Robust Command-Shaping Methods At the 1988 IEEE Conference on Robotics and Automation, Singer and Seering presented a paper on acausal command-shaping methods for controlling robot vibration.23 However, they gave only a brief overview of the paper and then used their remaining time to present a robust command-shaping method that they had recently developed.18,24,25 This robust method was a significant leap forward, as it greatly expanded the possible applications for command shaping. In a very short time, several other research groups adopted the idea and were making extensions and experimental verifications of Singer and Seering’s input-shaping method.26-31 In order to increase the robustness of the input-shaping process,

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Vtol

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Fig. 7 Input Shaper Sensitivity Curves Singer and Seering used an additional constraint to design their input shaper. Their constraint forced the derivative of the residual vibration, with respect to frequency, to equal zero: ∂ V (ω , ζ ) = 0. ∂ω

(16)

When (11), (12), (13), and (16) are satisfied with V(ω,ζ) = 0, the result is a Zero Vibration and Derivative (ZVD) shaper containing three impulses given by: 1 A 1 + 2 K + K 2 t = 0 i

i

K 1 + 2K + K 2 π ω 1−ζ 2

K2 1 + 2K + K 2 . 2π ω 1−ζ 2

(17)

By comparing the 5% insensitivities shown in Figure 7, it is obvious that the ZVD shaper is significantly more robust than the ZV shaper. Its 5% insensitivity is 0.286 – a 480% increase over the ZV shaper. Note that the cost of this robustness is a lengthening of the input shaper. The ZV shaper is 0.5 vibration periods in duration, while the ZVD shaper is one full period. This means that when the command is shaped, its rise time will be increased by one vibration period. This increase in rise time is usually a small price to pay for the robust vibration reduction. Singer and Seering proposed an extension to this idea where additional higher-order derivatives are formed and set equal to zero. When these additional constraints are used, the resulting shapers get more and more robust by further flattening the sensitivity curve at the modeling frequency. The cost for each additional robustness constraint is an additional lengthening of the input shaper by 0.5 vibration periods, and the need for one additional impulse in the input shaper. In his book5 that first described ZV shaping (posicast control), OJM Smith demonstrated that the shaper was effectively placing zeros over the flexible poles of the system – thereby canceling their vibratory effects. When derivative constraints are added to the problem formulation, the input shaper places additional zeros over the flexible poles of the plant.32,33 Soon after the development of the zero-derivative shapers, many other researchers sought to extend the robustness idea. The extensions can largely be categorized as: 1) Built-in robustness, or 2) Adaptive robustness. Built-in robustness seeks to make the input

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shaper inherently robust, as for example the ZVD shaper. A fundamental tradeoff in this approach is that obtaining more robustness leads to an increase in rise time. A key challenge is to obtain significant robustness with very little rise time penalty. Adaptive input shaping seeks to use feedback measurements of the system states to continually change the input shaper to improve its effectiveness. For example, the ZV shaper given in (14) can be continually changed during operation by updating the frequency, ω, that is used to calculate the shaper impulses. A significant cost of adaptive input shaping is that sensors must be added to the control system. Furthermore, these sensors must give indications of how the oscillation frequencies change. Key challenges in adaptive shaping are updating the shaper impulses rapidly and achieving stable behavior.

3.1 Built-in Robustness The zero derivative robustness constraints are only one possible way to improve the inherent robustness of input shapers. Soon after the ZVD shaper was disclosed, another approach was proposed in which the constraint of zero vibration at the modeling frequency was replaced with a constraint that merely limited the vibration to a small value.34-36 This approach is called the Extra-Insensitive (EI) approach because it provides extra robustness without increasing the shaper duration – it is the same duration as the ZVD shaper. The sensitivity curve for the EI shaper is shown in Figure 7. The nonzero vibration at the hump in the sensitivity curve, as well as the improved robustness, is readily apparent. Its 5% insensitivity is 0.4 – a 670% increase over the ZV shaper and a 140% increase over the ZVD shaper. As with the zero-derivative methods, the extrainsensitive method can be extended to higher levels of robustness by adding constraints that create more humps in the sensitivity curve.37 Another robust method, called Specified-Insensitivity (SI) shaping, suppresses a specified range of frequencies.38-40 The most straightforward method for generating a shaper with specified insensitivity to frequency errors is the technique of frequency sampling.38 This method requires repeated use of the vibration amplitude equation, (11). In each case, V(ω,ζ) is set less than or equal to a tolerable level of vibration, Vtol: V ≥ e−ζωs n t

tol

( C (ω , ζ ) ) + ( S (ω , ζ ) ) 2

s

s

2

, s = 1,..., m

(18)

where, ωs represents the m unique frequencies at which the vibration is limited. For example, if a frequency insensitivity of 0.4 is desired (±20% allowable error), then the constraint equations limit the vibration to below Vtol at frequencies between 0.8ωm and 1.2ωm. This procedure is illustrated in Figure 8 for Vtol = 5%. Given that the SI shapers can be designed to suppress any desired range of frequencies, they often prove advantageous over the ZVD and EI shapers. This flexibility in the robustness properties of the SI shapers is demonstrated in Figure 9, where the sensitivity curves for Iω = 0.5 and Iω = 0.7 are compared to that of the ZVD shaper. The SI design method can be extended by weighting the importance of the frequencies within the suppression range.41,42 (The method

Endpoint Vibration (%)

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shown in Figure 8 treats all frequencies with equal importance.) For example, the lower frequencies within the suppression zone could be weighted more – and suppressed more – because they are likely to produce larger amplitude residual vibration than higher frequencies. The SI design method can easily be extended to obtain robustness to modeling errors in the damping ratio. Constraints are simply added to the formulation to limit the residual vibration over a range of damping ratios. For example, Figure 10 shows the sensitivity curve for an SI shaper that was designed to suppress vibration over a range of frequencies from 0.7 Hz to 1.3 Hz, and also over a range of damping ratios from 0 to 0.2. Given the multiple choices of input shapers discussed here, (and many others not discussed), how does one choose which type of input shaper to use? A good approach is to analyze the properties of the various shapers and then choose the shaper that best fits the application in question. Input shapers have several properties of importance: Duration, Robustness, Ease of Implementation, HighMode Excitation, etc. Here we will review two of the most important properties: Robustness and Duration. Figure 11 shows the relationship between robustness and duration for several input shapers. The shaper duration is normalized by the vibration period. The SI shaper is plotted as a line because it can have any desired level of Insensitivity. The SI shaper has the minimum duration for any given Insensitivity. Therefore, SI shapers will provide the fastest rise time. One point of interest is that the EI shapers correspond to nodes on the SI shaper curve. This indicates that they offer the optimal insensitivity for their duration. It is also of interest to note that the zero-derivative method produces input shapers that provide substantially less Insensitivity than SI shapers.

1

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Fig. 11 Input Shaper Robustness vs. Shaper Duration

2

3.2 Adaptive Robustness Rather than construct an input shaper that has inherent robustness properties, many researchers have developed methods to use a non-robust shaper, but adapt its impulse amplitudes and time locations to the changing dynamic properties of the system. This approach can provide a faster rise time because the non-robust shaper is shorter in duration than a comparable robust shaper. Efforts on this approach appeared very soon after Singer and Seering’s ZVD shaper first appeared. One of the earliest adaptive input-shaping methods used a frequency-domain identification scheme to estimate the vibration frequencies and then update the spacing between the shaper impulses.28,29,43 The challenge with this approach is to perform the identification in real-time without placing too large of a computational burden on the control computer. Numerous experimental results were obtained to demonstrate the effectiveness of this approach.26,44,45 The approach can be modified to use other types of frequency identification methods, such as the Empirical Transfer Function Estimate approach.46 Given the time delay required to calculate the dominant mode from sensor data, one adaptive approach measured the second vibration mode and then used a known relationship between the first and second modes to calculate the primary mode.47 This

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method can be advantageous because the second mode can be calculated from fewer sensor measurements; therefore, it can be obtained faster than the first mode. The system identification can also be accomplished in the time domain.48,49 In the indirect adaptive approaches discussed above, the system parameters (natural frequency and damping ratio) are identified first and then the appropriate input shaper is designed. Another approach is to create a direct adaptation algorithm that never explicitly utilizes the system parameters. Instead, direct methods adapt the input shaper directly from the system output.50-52 In many cases, this approach can have better convergence characteristics than indirect approaches. Given the importance of the adaptation algorithm to the overall success of this approach to command shaping, it is important to compare the various methods and match their capabilities and properties with the application at hand.48,52-55 In addition to the real-time computational burden, another significant challenge for some applications is the effect of noise in the system. Noise can erroneously indicate that the dynamic properties are changing. This could lead to an incorrect change in the input shaper. This issue has been studied and a method to optimize solutions for systems with noise has been developed.56

against the period of the low mode. The horizontal axis shows the ratio between the high and low modes, R. For every mode ratio, the direct shapers have a smaller duration – they produce faster motion. Much of the work done in multi-mode command shaping has been directed at specific applications because the number of modes and their relative amplitudes and frequency ratios can significantly affect the design procedure that should be used. A number of papers devoted to multi-mode command shaping for specific applications such as robots,26,54,62-66 spacecraft,67-75 and cranes2,3,40,76-78 have been published. Some methods for multi-mode command shaping have sought to optimize, or satisfy, auxiliary constraints to improve certain aspects of the system performance. For example, an approach was developed to eliminate multiple modes with a minimum number of impulses in the input shaper.79 Using only a small number of impulses decreases the computational requirements during real-time implementation. Methods have also been developed to optimize the shaping process when multiple actuators are used to drive the system.80-84 Additional methods have been developed to optimally design command shaping to work in conjunction with feedback control85,86 and damping elements.87,88

4. Multi-mode Input Shapers

5. Input Shaping for Time-Optimal Control

Although many systems have one dominant flexible mode that causes most of the vibration problems, there are some systems where two or more modes must be addressed by the commandshaping method. Fortunately, command shaping is easily applied to multi-mode systems. Two basic approaches exist, 1) a convolution method wherein shapers designed for each of the modes are combined to create a multi-mode shaper and 2) a direct approach wherein constraints on all of the system frequencies are generated and then the full set of constraints are simultaneously solved to directly obtain a multi-mode shaper.25,57-59 The convolution design approach is easier to perform, but the direct, simultaneous approach can yield faster shapers with fewer impulses. The tradeoff between these two approaches has been thoroughly studied for two-mode systems.60,61 For example, Figure 12 compares the duration of convolved and direct ZVD shapers for two modes. The vertical axis shows the shaper duration, normalized

Time-optimal control for flexible systems is a special type of command shaping that seeks to create commands that will move a system as fast as possible from one state to another. This endeavor dates back to the 1960’s in the work of Pontryagin, but many other researchers followed on with contributions to this field.89-93 The approach requires a good model of the system dynamics, as well as knowledge of the actuator limits. The commands are then designed to use the maximum actuator effort to move the system as fast as possible. While the mathematical constructs for analyzing time-optimal commands for flexible systems were well developed, the actual calculation of such commands and their practical implementation lagged far behind. Such commands could only be generated for very simple systems and they suffered from poor robustness to modeling errors.94 These limitations were greatly reduced by the robustness concept developed by Singer and Seering. Wie and Liu were perhaps the first to see the possibility of using the zeroderivative robustness constraint in a time-optimal formulation that greatly reduces the sensitivity of time-optimal commands.95-97 The equivalence between time-optimal commands and input shaping occurs when the input shaper contains impulses whose amplitudes are [1, -2, 2, -2, …-2, 1]. Furthermore, this special type of input shaper must be convolved with a step input whose magnitude is equal to the maximum actuator force. This process is demonstrated in Figure 13. Rather than use the convolution process shown in Figure 13, Singh and Vadali demonstrated the timeoptimal input-shaping problem formulation and solution in the frequency domain.98 The extra-insensitive constraints were also formulated and solved for the special case of time-optimal control.99

Shaper Duration (Periods of Low Mode)

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produce on/off commands that are known in closed form.107-110 With these methods, the designer only has to enter the system dynamic properties (frequencies, actuator limits) and the desired move parameters (distance, fuel usage) into given equations and the switch times are immediately known. Such closed-form commands are also available for cases where the transient deflection111,112 and the fuel usage109,110 need to be limited. Another simple method for command shaping with on/off actuators is an approach where standard input-shaped commands are converted to an on/off function by approximation,113,114 or pulse-width modulation.115,116

Fig. 13 Input-Shaping Process to Generate Time-Optimal Commands

6. Example Applications

Max 0

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-1 -1 -1 -1 Input Shaper

Shaped Input -Max Fig. 14 Input Shaping Process to Generate Fuel-Efficient On/Off Commands. Given that the resulting time-optimal command is on/off, these commands can be used for on/off actuators such as reaction jets on spacecraft. Unfortunately, the time-optimal on/off commands are not efficient – they use a lot of actuator effort (fuel). This fact motivated several research groups to search for methods to make the commands more fuel efficient. Some of the methods start with an inherently fuel-efficient command profile and solve for the times at which it should switch on and off.100 Other methods use a weighting function between move speed and fuel usage,101-104 or simply allow the command designer to specify the amount of fuel that is to be used for any particular move.105 All of these methods can be formulated by simply changing the impulses magnitudes in the input shaper. For example, Figure 14 shows how an input shaper with amplitudes of ±1 can generate a command profile that has periods of coasting. This leads to a much more fuel-efficient command profile. The method shown in Figure 14 can produce on/off commands that move systems in a fuel-efficient manner (or with a specified amount of fuel); however, the transient deflection that the system undergoes is not directly controlled. Therefore, the system could be damaged during the motion by large internal deflection forces. This could be especially damaging for spacecraft, so researchers have modified the command-shaping process by adding additional constraints to limit transient deflection during the motion.47,106 The methods discussed so far in this section require a numerical optimization to solve the constraint equations and determine the impulse times (command switch times). This can lead to large computational burdens that hamper real-time implementation. To alleviate this problem, researchers have developed methods that

Given the ease with which command shaping can be implemented, and its robust effectiveness, it is not surprising that it has been implemented on millions of machines throughout the world. The responses of the bridge crane that were shown in Figure 3 demonstrate the benefit of command shaping crane commands. In addition to the single-pendulum bridge crane shown in Figure 3, command shaping has been implemented on tower cranes,117 boom cranes,118,119 container cranes,78,120-123 double-pendulum cranes,40 and cranes whose payloads bounce when they are lowered.124 While crane oscillation is large-amplitude and low-frequency in nature, command shaping also works well on high-frequency, lowamplitude vibration. Consider the moving-bridge coordinate measuring machine (CMM) sketched in Figure 15. The machine is composed of stiff components including a granite base and largecross-section structural members. The goal of the CMM is to move a measurement probe throughout its workspace so that it can contact manufactured parts that are being inspected. In this way, it can accurately determine the dimensions of the part and ensure that they meet specifications. The position of the probe is measured by optical encoders that are attached to the granite base and the moving bridge. However, the measured location of the probe will differ from the actual location because the physical structure deflects between the encoders and the probe endpoint.

Measured Part

TouchTrigger Probe

Fig. 15 Moving-Bridge Coordinate Measuring Machine

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Unshaped Response Shaped Response

40

Gimbal Position (degrees)

Deflection (Laser-Encoder,µm)

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20 0.0 -20 Measurement

-40 -60 0.40

0.60

0.80 1.00 Time(sec)

1.20

Fig. 16 Comparison of Deflections During Shaped and Unshaped Measurement Cycles When command shaping is applied to the reference command of a CMM, the deflection of the structure is reduced, both during the gross motion and during the slow approach toward the part.125127 This effect is shown in Figure 16. The command shaping reduced the measurement probe deflection from approximately 25 µm to about 6 µm during the critical measurement phase. Successful applications of input shaping have also been reported on the related problems of controlling XY stages,128-133 flexible robotic arms,18,28,51,62,65,66,68,134-156 and voice coil motors.157 Implementation on these types of industrial machines may require the input-shaping algorithm to deal with friction,116,158,159 saturation,75,160 or rate limiting.81,161-165 Command shaping has also proven beneficial for spacecraft.67,69-71,73-75,95,97,99,100,106,109,115,166-170 Input shapers were tested in space using the Middeck Active Control Experiment (MACE).37,171,172 The MACE hardware was designed to represent a typical satellite with multiple pointing mechanisms. This experimental apparatus first flew on board the Space Shuttle Endeavor in March 1995, as shown in Figure 17. It was then relaunched in September of 2000 to the International Space Station. A goal of the experimental program was to develop control algorithms that would allow gimbals on both ends of the structure to simultaneously make accurate motions. This requires each gimbal to be robust to disturbances caused by the motion of the other gimbal. Figure 18 shows the shaped and unshaped responses of the

1.5 1 0.5

Unshaped Step Shaped Step

0 -0.5 -1 -1.5

0

1

2

3

4

5

Time (sec)

6

7

8

Fig. 18 MACE Step Responses primary gimbal when it was moved approximately 3 degrees. Command shaping eliminated virtually all of the residual vibration. The very low frequency drift in the position was caused by the umbilical that connected the free-floating hardware to the space shuttle.

7. Conclusions Methods that generate specially-shaped reference commands to move flexible systems are an important part of control theory and application – over 700 papers on the subject have been published. The major advantage of such commands is that they do not require sensor measurements; although sensors can be used in adaptive command-shaping methods to improve performance. Another strength of command shaping is that it acts to suppress vibration in a preemptive way that is faster than anything possible with feedback control. Feedback control must wait for an error to arise and be sensed before it starts to suppress it. Command shaping uses a dynamic model to anticipate the occurrence of vibration, so it can effectively start to act as soon as the system starts to move. A key advancement in command shaping was the concept of robustness – commands can be designed to work well, even when large modeling errors exist. Furthermore, the required dynamic model is usually quite simple – just estimates of the natural frequencies and damping ratios. Command shaping is very versatile in that many types of auxiliary constraints, such as actuator limits, fuel usage, and transient deflection limits, can be integrated into the design of the commands. These beneficial properties have enabled engineers to harness the benefits of command shaping on millions of machines around the world.

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Fig. 17 MACE Experiment Onboard the Space Shuttle. Photo courtesy of NASA

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130. deRoover, D., “Motion Control of a Wafer Stage,” Delft University Press, 1997. 131. Park, S.-W., Hong, S.-W., Choi, H.-S. and Singhose, W., “Discretization Effects of Real-Time Input Shaping in Residual Vibration Reduction for Precise XY Stage,” Trans. of the Korean Society of Machine Tool Engineers, Vol. 16, pp. 71-78, 2007. 132. Vaughan, J., Yano, A. and Singhose, W., “Comparison of Robust Input Shapers,” Journal of Sound and Vibration, Vol. 315, pp. 797-815, 2008. 133. Vaughan, J., Yano, A. and Singhose, W., “Robust Negative Input Shapers for Vibration Suppression,” ASME J. Dynamic Systems, Measurement, and Control, Vol. 131, pp. 031014-1-9, 2009. 134. Starr, G. P., “Swing-Free Transport of Suspended Objects with a Path-Controlled Robot Manipulator,” J. of Dynamic Systems, Measurement and Control, Vol. 107, pp. 97-100, 1985. 135. Murphy, B. R. and Watanabe, I., “Digital Shaping Filters for Reducing Machine Vibration,” IEEE Transactions on Robotics and Automation, Vol. 8, pp. 285-289, 1992. 136. Feddema, J. T., “Digital Filter Control of Remotely Operated Flexible Robotic Structures,” Proc. American Control Conf., pp. 2710-2715, 1993. 137. Yano, K. and Terashima, K., “Sloshing suppression control of liquid transfer systems considering a 3-D transfer path,” IEEE-ASME Transactions on Mechatronics, Vol. 10, pp. 8-16, 2005.

124. Yoon, J., Singhose, W., Kim, M. D., Ramirez, G. and Tawde, S. K., “Dynamics and Control of Bouncing and Tilting Crane Payloads,” ASME IDETC, 2009.

138. Feddema, J., Dohrmann, C., Parker, G., Robinett, R., Romero, V. and Schmitt, D., “Control for Slosh-Free Motion of an Open Container,” IEEE Control Systems, Vol. 17, pp. 29-36, 1997.

125. Jones, S. and Ulsoy, A. G., “An Approach to Control Input Shaping with Application to Coordinate Measuring Machines,” J. of Dynamics, Measurement, and Control, Vol. 121, pp. 242-247, 1999.

139. Zou, K., Drapeau, V. and Wang, D., “Closed Loop ShapedInput Strategies for Flexible Robots,” Int. J. of Robotics Research, Vol. 14, pp. 510-529, 1995.

126. Seth, N., Rattan, K. and Brandstetter, R., “Vibration Control of a Coordinate Measuring Machine,” Proc. IEEE Conf. on Control Apps., pp. 368-373, 1993.

140. Rappole, B. W., Singer, N. C. and Seering, W. P., “MultipleMode Impulse Shaping Sequences for Reducing Residual Vibrations,” Proc. 23rd Biennial Mechanisms Conference, pp. 11-16, 1994.

127. Singhose, W., Singer, N. and Seering, W., “Improving Repeatability of Coordinate Measuring Machines with Shaped Command Signals,” Precision Engineering, Vol. 18, pp. 138146, 1996. 128. Fortgang, J., Singhose, W. and Márquez, J. d. J., “Command Shaping for Micro-Mills and CNC Controllers,” Proc. American Control Conference, 2005. 129. Singhose, W. and Singer, N., “Effects of Input Shaping on Two-Dimensional Trajectory Following,” IEEE Trans. on Robotics and Automation, Vol. 12, pp. 881-887, 1996.

141. Kwon, D.-S., Hwang, D.-H., Babcock, S. M. and Burks, B. L., “Input Shaping Filter Methods for the Control of Structurally Flexible, Long-Reach Manipulators,” Proc. IEEE Conf. on Robotics and Automation, pp. 3259-3264, 1994. 142. Chang, T., Godbole, K. and Hou, E., “Optimal input shaper design for high-speed robotic workcells,” Journal of Vibration and Control, Vol. 9, pp. 1359-1376, 2003. 143. Grosser, K. and Singhose, W., “Command Generation for Reducing Perceived Lag in Flexible Telerobotic Arms,” JSME

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International Journal, Vol. 43, pp. 755-761, 2000. 144. Wilson, D. G., Stokes, D., Starr, G. and Robinett, R. D., “Optimized Input Shaping for a Single Flexible Robot Link,” SPACE 96: 5th International Conf. and Expo. on Engineering, Construction, and Operations in Space, 1996. 145. Singhose, W., “Trajectory Planning for Flexible Robots: CRC Robotics and Automation Handbook, T. Kurfess, Ed.,” CRC Press, 2004. 146. Mohamed, Z. and Tokhi, M. O., “Command Shaping Techniques for Vibration Control of a Flexible Robot Manipulator,” Mechatronics, Vol. 14, pp. 69-90, 2004. 147. Alici, G. U., Kapucu, S. and Baysec, S., “A Robust Motion Design Technique for Flexible-Jointed Manipulation Systems,” Robotica, Vol. 24, pp. 95-103, 2006. 148. Chang, T., Jaroonsiriphan, P., Bernhardt, M. and Ludden, P., “Web-based command shaping of cobra 600 robot with a swinging load,” IEEE Transactions on Industrial Informatics, Vol. 2, pp. 59-69, 2006. 149. Freese, M., Fukushima, E., Hirose, S. and Singhose, W., “Endpoint Vibration Control of a Mobile Mine-Detecting Robotic Manipulator,” Proc. American Control Conference, 2007. 150. Gurleyuk, S. S., “Optimal unity-magnitude input shaper duration analysis,” Archive of Applied Mechanics, Vol. 77, pp. 63-71, 2007. 151. Huey, J. R., Sorensen, K. L. and Singhose, W. E., “Useful Applications of Closed-Loop Signal Shaping Controllers,” Control Engineering Practice, Vol. 16, pp. 836-846, 2008.

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157. Jung, J.-K., Youm, W.-S. and Park, K.-H., “Vibration Reduction Control of a Voice Coil Motor (VCM) Nano Scanner,” IJPEM, Vol. 10, No. 3, pp. 167-170, 2009. 158. Rathbun, D. B., Berg, M. C. and Buffinton, K. W., “Pulse width control for precise positioning of structurally flexible systems subject to stiction and coulomb friction,” J. of Dynamic Systems, Measurement and Control, Vol. 126, pp. 131-138, 2004. 159. Lawrence, J., Singhose, W. and Hekman, K., “FrictionCompensating Input Shaping for Vibration Reduction,” ASME J. of Vibration and Acoustics, Vol. 127, pp. 307-314, 2005. 160. Robertson, M. J. and Erwin, R., “Command Shapers for Systems with Actuator Saturation,” Proc. American Control Conference, pp. 760-765, 2007. 161. Andersch, P., Sorensen, K. and Singhose, W., “Effects of Rate Limiting on Common Input Shaping Filters,” Recent Advances in Systems Engineering and Applied Mathematics, pp. 33-38, 2008. 162. Meckl, P. H., Arestides, P. B. and Woods, M. C., “Optimized S-Curve Motion Profiles for Minimum Residual Vibration,” Proc. American Control Conference, pp. 2627-2631, 1998. 163. Singh, T., “Jerk Limited Input Shapers,” J. of Dynamic Systems, Measurement, and Controls, Vol. 126, pp. 215-219, 2004. 164. Danielson, J., Lawrence, J. and Singhose, W., “Command Shaping for Flexible Systems Subject to Constant Acceleration Limits,” ASME J. of Dynamic Systems, Meas., and Control, Vol. 130, pp. 0510111-0510118, 2008.

152. Kinceler, R. and Meckl, P. H., “Corrective Input Shaping for a Flexible-joint Manipulator,” Proc. American Control Conference, pp. 1335-1339, 1997.

165. Pao, L. Y. and La-Orpacharapan, C., “Shaped Time-Optimal Feedback Controllers for Flexible Structures,” J. Dynamic Systems, Measurement, and Control, Vol. 126, pp. 173-186, 2004.

153. Kozak, K., Singhose, W. and Ebert-Uphoff, I., “Performance Measures For Input Shaping and Command Generation,” ASME J. Dynamic Systems, Meas., & Control, Vol. 128, pp. 731-736, 2006.

166. Banerjee, A., Singhose, W. and Blackburn, D., “Orbit Boosting of an Electrodynamic Tethered Satellite with InputShaped Current,” Proc. AAS/AIAA Astrodynamics Specialist Conference, 2005.

154. Parker, G. G., Eisler, G. R., Phelan, J. and Robinett, R. D., “Input Shaping for Vibration-Damped Slewing of a Flexible Beam Using a Heavy-Lift Hydraulic Robot,” Proc. American Control Conf., 1994.

167. Parman, S. and Koguchi, H., “Controlling the attitude maneuvers of flexible spacecraft by using time-optimal/fuelefficient shaped inputs,” Journal of Sound and Vibration, Vol. 221, pp. 545-565, 1999.

155. Tokhi, M. O. and Mohamed, Z., “Combined Input Shaping and Feedback Control of a Flexible Manipulator,” 10th International Congress on Sound and Vibration, pp. 299-306, 2003.

168. Gorinevsky, D. and Vukovich, G., “Nonlinear Input Shaping Control of Flexible Spacecraft Reorientation Maneuver,” J. of Guidance, Control, and Dynamics, Vol. 21, pp. 264-270, 1998.

156. Sung, Y.-G. and Lee, K.-T., “An Adaptive Tracking Controller for Vibration Reduction of Flexible Manipulator,” IJPEM, Vol. 7, No. 3, pp. 51-55, 2006.

169. Biediger, E., Singhose, W., Okada, H. and Matunaga, S., “Trajectory Planning for Coordinating Satellites using Command Generation,” Proc. 10th International Space Conference of Pacific-basin Societies, 2003.

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170. Singhose, W., Biediger, E., Okada, H. and Matunaga, S., “Experimental Verification of Real-Time Control for Flexible Systems with On-Off Actuators,” ASME J. of Dynamic Systems, Measurement, and Controls, Vol. 128, pp. 287-296, 2006. 171. Ninneman, R. and Denoyer, K., “Middeck Active Control Experiment Reflight (MACE II): Lessons learned and reflight status,” Proc. of SPIE - The International Society for Optical Engineering, pp. 131-137, 2000. 172. Tuttle, T. and Seering, W., “Experimental Verification of Vibration Reduction in Flexible Spacecraft Using Input Shaping,” J. of Guidance, Control, and Dynamics, Vol. 20, pp. 658-664, 1997.

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