Input shaping is widely used in the control of flexible systems due to its effectiveness and ease of implementation. Due to its open-loop nature, it is often overlooked as a control method in systems where parametric uncertainty or force disturbances are present. However, if the disturbances are known and finite in duration, their effect on the flexible mode can be approximated by formulating an initial condition control problem. With this knowledge, an input shaper can be designed, which cancels the initial oscillation, resulting in minimal residual vibration. By incorporating Specified Insensitivity robustness constraints, such shapers can be designed to ensure good performance in the presence of modeling uncertainty. This input shaping method is demonstrated through computer and experimental methods to eliminate vibration in actuator bandwidth-limited systems.

## Introduction

Flexible mechanical systems offer a number of benefits over their rigid counterparts. Because flexible structures are inherently lighter, they require lower actuator effort for quick point-to-point motion [1]. In turn, they are more time and energy efficient compared to bulky, rigid manipulators. These benefits are only relevant if the vibration in the flexible system can be controlled. To this end, a large number of vibration reduction methods can be employed.

Open-loop control methods have been employed in a large range of systems to reduce vibration resulting from point-to-point motion. One particular open-loop technique, noted for its simplicity and ease of implementation, is the command shaping method called input shaping [2]. This method, first introduced as “Posicast Control,” generates a sequence of impulses called an input shaper, which is convolved with a reference command to produce a new, shaped command that results in significantly reduced vibration [3]. Input shaping has been used to control a variety of flexible systems including coordinate measurement machines [4] and cranes [5,6]. This technique has also been employed to compensate for nonlinearities due to friction [7], multimodal systems [6,8], limited actuator bandwidth [9], and perceived overshoot in human-operated systems [10].

The open-loop nature of input shaping renders it incapable of rejecting oscillation resulting from external disturbances or parametric uncertainty. To compensate for uncertain plant dynamics, robust input shapers, which result in acceptably low levels of vibration for a range of frequencies, can be designed [11,12]. If disturbance rejection is desired, an input shaper can be used in conjunction with feedback control [13,14].

In some cases, disturbances introduced to a system can be approximated by an impulse, which causes nonzero initial conditions. When this is true, a series of impulses can be designed, which eliminates the initial oscillation [15–21]. This approach has been proposed to reject vibration due to a change in the desired setpoint of a time-optimal trajectory for a flexible system [19], a known force on a long-reach telescopic handler [18], the transient sway of a harmonically excited boom crane [16], and the point-to-point motion of cranes at nonzero initial conditions [15,20,21]. The general procedure for constructing these series of impulses is called initial condition input shaping.

In addition to the previously mentioned applications, initial condition input shaping has a number of potential uses in the control of flexible systems. For example, the slewing motion of a boom or tower crane introduces radial and tangential swing [22]. The input shaping method, which cancels payload swing in both directions, can be formulated as an initial condition input shaper. Another potential application of the initial condition input shaping method is in the hoist of off-centered payloads [23]. If the payload deflection at the time of hoisting is known, a shaped command can be generated, which brings the payload to rest [15,21,20].

This command shaping method requires accurate knowledge of the plant dynamics as well as vibration-inducing forces. Although a number of widely used closed-loop methods such as sliding-mode control or linear quadratic regulation could be employed to accomplish the same task, initial condition input shaping presents several unique characteristics, which make it worthy of consideration. In the given crane examples, a single position measurement alongside knowledge of the plant dynamics provides enough information to eliminate the undesired vibration. By reducing the reliance on potentially noisy sensor data, the controller can be simplified [24]. This benefit is most clearly demonstrated in the given examples, where the initial-condition-causing force is known and not expected to persist.

This work will present a thorough analysis on the design and implementation of initial condition input shapers for vibration suppression. This method is based on the previously cited literature, which develops impulse-based control in a similar manner. However, this paper presents a more rigorous theoretical development and directly applies common input shaping tools to yield a generalizable solution. Furthermore, moderate nonlinearities due to actuator bandwidth constraints and system dynamics are directly addressed in order to yield better vibration reduction. Finally, robustness to modeling errors is addressed by implementing the Specified Insensitivity input shaping technique. Using the approach presented in this paper, initial-condition-canceling input shapers with an arbitrary level of robustness can be developed.

The paper is organized as follows: Section 2 will provide relevant background on the input shaping process. Section 3 develops the solution for initial condition shapers through frequency and time domain analysis. Robustness considerations and actuator limitations are considered in this section as well. Next, example responses are presented in Sec. 4. Section 5 provides a demonstration of initial condition input shaping the luff of a planar boom crane. The shaping methods are experimentally validated in Sec. 6. Conclusions are given in Sec. 7.

## Input Shaping Overview

*n*impulses normalized by a unity magnitude impulse at time

*t*= 0

*A*and

_{i}*t*are the

_{i}*i*th impulse amplitudes and times,

*ω*is the natural frequency, and

*ζ*is the damping ratio. Setting (1) equal to zero for a given

*ω*and

*ζ*is exactly equivalent to pole cancellation in the frequency domain. If a more robust solution is desired, derivatives of (1) with respect to

*ω*can be set to zero [25], resulting in repeated zeros at the system poles [26]. Because some modeling error is inevitable, relaxing the vibration constraint for a range of sampled frequencies

*V*

_{tol}[27]. This generalized vibration constraint refers to the Specified Insensitivity method, where closed-form solutions are available for special cases and are referred to as extra-insensitive shapers [28]. The numerous methods of improving robustness for input shapers are summarized in Ref. [11].

where the first constraint ensures that no impulse exceeds magnitude of 1, and the second constraint forces the cumulative sum of any number of impulses to not exceed the desired set-point.

## Initial Condition Input Shaping

If a flexible system exhibits nonzero initial states, an input shaper can be designed, which approximates the initial states as an additional impulse to be canceled by the shaper impulse sequence. This process is demonstrated graphically on a vector diagram [19,27] such as Fig. 2. Here, the initial condition is modeled as an impulse of magnitude *A*_{0} at phase $\theta 0=\omega dt0$. The input shaper impulses *A*_{1} and *A*_{2} sum to produce a resultant impulse *A _{s}*, which is equal in magnitude and directly out of phase with

*A*

_{0}, resulting in zero residual vibration.

### Frequency-Domain Design.

Here, *y*_{0} and $y\u02d90$ are the displacement and velocity of the flexible mode, and $t1=0$.

where $s0=\u2212\zeta \omega \xb1j\omega 1\u2212\zeta 2$.

*t*

_{2,}which satisfies the constraint that the imaginary component of (12) be zero. For the undamped case, the solution is [17]

### Time-Domain Design.

*t*= 0, the effects of the initial condition must be incorporated into the normalization term. Specifically

*A*

_{0}while maintaining the desired robustness. As a result, robust input shapers, which cancel a wide range of initial conditions, can be generated. The expression for a SI-IC shaper is therefore

Although the robust SI-IC shaper solution is presented here in the time domain, an analogous formulation could be presented in the frequency domain through minimax optimization [29].

### Incorporating Actuator Limitations.

In the previous analysis, it is assumed that the actuators can exactly follow the impulse commands from the input shaper. Obviously, no actuator can exert an infinite force, so pulses of finite amplitude must be used. Although this distinction is unnecessary for typical input shaping implementation, the differences between the impulse and pulse responses must be quantified for an initial-condition-canceling input shaper.

Figure 3 demonstrates the phase and amplitude shifts, $\varphi $ and *δ*, respectively, of a pulse response compared to an impulse response. These shifts affect the performance of an IC shaper as demonstrated on the vector diagram in Fig. 4. The phase shift, $\varphi $, corresponds to an apparent lag in the resulting input shaper impulse while the amplitude shift, *δ*, corresponds to the reduced vector amplitude.

*τ*, then these relationships can be determined by iteratively fitting a shifted response

_{d}*τ*

_{acc}, and damping ratio,

*ζ*. In Fig. 5, a linear relationship between $\varphi $ and

*τ*

_{acc}is evident for each

*ζ*value. This slope increases as a function of

*ζ*. Figure 6 shows that for low damping levels, the amplitude shift trends toward zero as

*τ*

_{acc}increases. Conversely,

*δ*becomes extremely large when both

*ζ*and

*τ*

_{acc}are also high. The data in this figure are clipped at

*δ*= 5 to improve clarity. This trend toward high

*δ*values occurs because the phase shift of the pulse command delays the response beyond the point at which the impulse response is settled.

### Robustness Considerations.

*t*= 0. For a system at rest, this expression simplifies to Eq. (1). Because the system under consideration exhibits nonzero initial conditions, the sensitivity curve takes a form similar to Eq. (25). Shaper performance in this case will further be degraded by inaccurately timing the shaped command or designing for an incorrect initial condition amplitude. Therefore, the full expression of the sensitivity of the input shaper is

*A _{e}* and

*θ*are errors in the designed shaper amplitude and phase, respectively. Equations (32) and (33) represent the oscillation amplitude due to the initial condition subject to these measurement errors. An initial condition input shaper can be made robust to deviations in

_{e}*ω*and

*ζ*, but not

*A*and

_{e}*θ*. However, quantifying the degradation of shaper performance subject to these errors provides valuable insight into how accurate the state estimation must be for this technique to function properly.

_{e}Because Eq. (31) is normalized by the response amplitude of a unity magnitude impulse at *t* = 0, subject to nonzero initial conditions, the exact shape of the shaper sensitivity curve will depend on the initial conditions. However, the sensitivity function is smooth and continuous near the constraints. Therefore, shaper performance around the design parameters is similar for a variety of initial conditions.

Regardless of the enforced robustness constraints, IC shapers exhibit nearly identical sensitivity to initial conditions. This performance measure can be quantified on a sensitivity plot such as the one shown in Fig. 7. Here, perfect modeling of the system dynamics, *ω* and *ζ*, is assumed. The residual vibration is zero at exactly one point, and quickly degrades as errors in initial condition phase and amplitude increase. For a shaper with specified insensitivity constraints, this curve will be shifted slightly off-center due to the nonzero vibration resulting at the designed conditions.

Figure 8 demonstrates the combined sensitivities of a ZV-IC shaper to initial conditions and normalized frequency error. Because the shaper is designed to exhibit zero residual vibration at the modeled frequency, a single point exists, which minimizes this sensitivity along each dimension. Taking a cross section normal to the $\omega n/\omega m$ axis for each plot results in a standard sensitivity curve for the given initial condition values. The insensitivity of an SI-IC shaper is significantly different, as shown in Fig. 9, due to the substantially increased robustness along the $\omega n/\omega m$ axis.

## Impulse Response Example

*I*= 0.4. The resulting input shapers are

Figure 10 shows the response of the system subject to a ZV-IC-shaped pulse input. As expected, the shaped command completely eliminates the residual vibration. A shaped command that was designed without consideration to the actuator limitations is also given for comparison. The importance of the phase and amplitude shifts resulting from these limitations is evident in the residual vibration resulting from the unshifted command. Finally, a modeling error of $(\omega n/\omega m)=1.2$ is introduced to demonstrate the effect of such an error on the shaped response. Because the ZV-IC shaper does not incorporate any robustness constraints, the performance suffers as a result of this change.

Similarly, the SI-IC-shaped responses are shown in Fig. 11. Because a tolerable level of vibration is permitted at the design frequency, a small amount of residual vibration exists after completing the shaped command. Furthermore, the robustness constraints result in a longer duration of the shaped command. This robustness is evident in the response subject to the modeling error of $(\omega n/\omega m)=1.2$, the upper limit of the suppressed frequency range. The residual vibration subject to this error is significantly lower than the ZV-IC-shaped case. The response of the system subject to an SI-IC shaper designed without considering the actuator limitations results in an increased level of vibration, as expected.

The robustness of these shapers to modeling uncertainty is given by the sensitivity curve in Fig. 12. While the ZV-IC shaper is designed to yield no residual vibration at the designed frequency, it rapidly loses effectiveness when the actual natural frequency deviates from this value. The SI-IC shaper, on the other hand, maintains a low level of vibration across the entire range of frequencies. This plot is consistent with the previous simulation responses.

## An Application in Crane Control

The proposed command shaping method can be applied to eliminate nonzero initial states in a weakly nonlinear system such as a boom crane as shown in Fig. 13 [15]. This simple model is composed of a rigid, massless boom and cable of lengths *R* and *l*, respectively. Payload *m* oscillates about point B by swing angle $\varphi $, and the control input is the boom luff angle, *γ*.

In order to design a shaper, which eliminates the nonzero initial states $\varphi 0$ and $\varphi \u02d90$, the pulse response must be once again compared to the impulse response for an analogous linear system. The approach for the boom crane is nearly identical to that of a linear damped, second-order system.

*t*

_{acc}, the swing response subject to a luff command can be approximately characterized by a linear system with

where *γ*_{0} is the initial luff angle. To normalize the initial conditions of the boom crane, the commanded pulse is scaled by the radial velocity of point *B* at the beginning of the command. This scaling factor is configuration dependent, based on the initial luff angle. A boom crane exhibits slightly different dynamics from a simple linear system after this consideration. Therefore, a second amplitude shift, *δ _{bc}* and phase shift, $\varphi bc$ must be computed and used to shift the target initial conditions in the same way as demonstrated in Sec. 3.3. Once the initial conditions are shifted in this manner, the ZV-IC and SI-IC shaper can be solved using the procedures presented in this work.

A representative simulated response is shown in Fig. 14. The boom crane begins at nonzero initial conditions and undergoes a shaped luff command. The $bang\u2212on$ is shaped by the IC shapers while the $bang\u2212off$ portion of the command is shaped by a zero vibration and derivative shaper [25] to clearly demonstrate the performance of each IC shaper. The IC shapers are designed to cancel the nonzero initial conditions while performing an upward luff command. The moderate nonlinearity of the boom crane results in a small level of oscillation in the ZV-IC response, as is expected. The SI-IC shaper results in slightly more residual vibration.

## Experimental Verification

To further validate this command shaping method, an experimental platform at the Kumoh National Institute of Technology in Korea, pictured in Fig. 15, was used. In this system, the flexible rod is commanded to move along the track, resulting in oscillation of the mass. This oscillation is measured by a laser scan micrometer.

The values used in this experimental analysis are summarized in Table 2. The natural frequency and damping ratio were determined experimentally by measuring the free response of the system for a specified mass height. The acceleration time, *t*_{acc}, and maximum velocity, *V*_{max}, were determined by analyzing the step response. Note that the location of the laser micrometer is near the base of the flexible rod. As a result, the measured deflection is a scaled-down approximation of the actual payload deflection. Although the flexible beam system is multimodal, the lowest mode, *ω _{n}*, dominates the response. Therefore, the measurement acquired at this point serves as an accurate proxy for the deflection of the tip.

A typical system response is presented in Fig. 16. Here, the experimental response is compared to simulation predictions based on a linear model. A known impulse force generates oscillation, represented by the position and velocity of the flexible mode, prior to the beginning of the command. In this unshaped case, the command-induced vibration results in a higher level of oscillation than what existed as a result of the initial conditions. The residual vibration is measured by the amplitude of oscillation after the completion of the commanded motion. This amplitude will be used to compare the performance of the shapers to the unshaped case. In order to minimize experimental error in this analysis, the end of the command is shaped using a standard ZV shaper based on the calculated natural frequency and damping ratio of the system. This ZV shaper introduces approximately no additional vibration into the system while bringing the rigid mode to rest, allowing for consistent measurement of the residual vibration due to the shaping methods under consideration.

The ZV-IC and SI-IC input shapers were designed to eliminate the given initial conditions. Their responses at the designed natural frequency are shown in Figs. 17 and 18. The more complex impulse sequence of the SI-IC shaper yields higher transient vibration while the system completes the motion, but both shapers result in low levels of residual vibration. In each case, the experimental trials also closely resemble the simulated results.

Both shaping methods were tested for robustness to modeling uncertainty by determining the residual vibration amplitude of the shaped and unshaped responses at various natural frequencies. The results for the ZV-IC shaper are summarized in Fig. 19(a). This plot compares the theoretical residual vibration amplitude subject to deviations in natural frequency, given by (31), to those found by experimental trials. Each experimental data point in this figure is the mean of three trials, where the variance between each trial is approximately zero. Here, the data closely match the predicted values, particularly near the designed natural frequency. Because the natural frequency of the experimental system is experimentally estimated and assumed to vary linearly, some modeling error due to nonlinear dynamics of the system is to be expected. Note that in this plot, the residual vibration remains below the unshaped case for all sampled frequencies.

Similar results for the SI-IC shaper are shown in Fig. 19(b). This shaper was designed to suppress vibration in a range of frequencies, $[0.9\omega ,1.1\omega ]$. A trend similar to the ZV-IC results is evident for these trials; the data increasingly deviate from the theoretical prediction at frequencies significantly different from the modeled frequency. In the suppressed range, however, the experimental data show that the shaper correctly minimized residual vibration. As a result of the negative amplitudes in this shaper, the residual vibration percentage increases more quickly as the natural frequency deviates from the suppressed range. The shaped command results in greater residual vibration than the unshaped command at greater than approximately 20% natural frequency error.

Figure 20 provides additional insight into the performance of the shaped commands relative to the unshaped command. This plot shows the residual vibration amplitude of the shaped commands as well as the unshaped command. An approximately linear decrease in vibration levels at higher frequencies is visible for the unshaped command. Because the residual vibration is measured at the end of the command, the effects of damping are apparent in this vibration amplitude measurement.

While the natural frequency increases and the command duration is constant, more oscillation is damped during the command, resulting in this measurement trend. Although the effect of this damping is significant on the measured residual vibration amplitudes of each command shaping method, all three are affected approximately equally.

## Conclusions

This work has introduced multiple methods of designing input shapers, which are capable of eliminating initial oscillation in a flexible system. The frequency domain solution can be solved in closed form for an undamped system, while it requires a simple optimization for the more general, damped case. A time domain design procedure can be used to generate IC shapers, which are robust to modeling uncertainty. Additionally, these shapers can be modified based on the actuator constraints of the system. Experimental results validated the proposed shaping methods. The experimental system responses closely matched those predicted by simulation. Furthermore, the robustness of the ZV-IC and SI-IC shapers was measured subject to varying natural frequencies. These experimental results support the effectiveness of each shaping method, while demonstrating the increased robustness of the SI-IC shaping approach.

## Funding data

Louisiana Board of Regents and ASV Global (LEQSF(2014-17)-RD-B).

National Science Foundation and the Korean National Research Foundation for providing funding through the East Asia Pacific Summer Institutes (EAPSI) Program (1714041).