Technical notes on input shaping in overhead crane systems

Introduction

Input shaping has evolved from the basic Zero-Vibration (ZV) shaper to more robust variants like Zero-Vibration and Derivative (ZVD) and Extra-Insensitive (EI) shapers. Multi-mode, constrained, and adaptive versions further expand its applicability to complex crane systems. Hybrid strategies integrate multiple input shapers within a single payload transport maneuver to adapt to varying dynamic conditions and enhance overall performance and timing.^1–6

However, several mathematically arguable issues in developing input shaping strategies, ranging from terminological clarity to sensitivity to parameter uncertainties and the handling of complex dynamic models, warrant in-depth analytical discussion.

Furthermore, power consumption, actuator effort, and human factors remain underexplored in the design of input shaping strategies. Moreover, the lack of standardized benchmarks and performance indices limits the systematic evaluation and comparison of existing methods. Addressing these gaps requires the development of comprehensive performance metrics, robust operation under operator interaction and potential faults, and a unified framework for evaluating input shaping techniques in practical crane applications.

This article discusses the principles, design strategies, and modeling challenges of input shaping, highlighting subtle aspects relevant to overhead crane systems.

Vibration versus oscillation

Several papers on overhead cranes use the term “vibration” to describe the motion of the payload. However, this terminology can be misleading, as “vibration” typically refers to high-frequency, often small-amplitude motions. In contrast, the payload motion in overhead cranes is predominantly characterized by low-frequency, pendulum-like swings induced by trolley acceleration and deceleration. Therefore, the term “oscillation” is more appropriate and descriptively accurate in this context, as it better captures the nature of the small, low-frequency swinging motion of the payload.

Input functions

The crane-trolley input functions are of significant importance due to their substantial influence on system performance. Their role extends beyond merely smoothing the input profile to fundamentally shaping the mathematical formulation of the control strategy. Traditional input shaping methods rely on a sequence of idealized impulse signals. However, this approach is both impractical and undesirable, as it introduces abrupt shocks and impacts into the mechanical system, potentially leading to severe long-term issues.

Moreover, ideal impulses, defined as signals with nearly zero-time duration and exact amplitude, are physically unrealizable as no mechanical input force can be shaped in nearly zero time. Moreover, in practice, any deviation from these ideal conditions results in shaping inaccuracies, particularly for sensitive shapers such as the ZV shaper. These inaccuracies, coupled with the inherent dynamic characteristics of mechanical systems, such as delayed responses and limitations in tracking abrupt input changes, make it difficult for actuators, such as the driving motor, to accurately follow the commanded input. This mismatch degrades system performance and destabilizes the effectiveness of impulse-based input shaping.

Nevertheless, alternative shaping functions can replace conventional impulse signals. Multiple step inputs, for instance, may serve as a semi-practical substitute. However, sensitive or multi-objective shapers often require a considerable number of input steps to satisfy the system conditions and performance criteria. These step-based profiles still exhibit abrupt transitions that actuators cannot precisely track, especially in systems with limited bandwidth or dynamic constraints. As a result, such approaches remain unsuitable for sensitive applications where accuracy and smoothness of control are critical.

Harmonic functions have been employed in various studies as alternative input shaping methods. Simple harmonic models consisting of two or three low-frequency terms are often effective and preferable, particularly due to their smooth profiles and reduced excitation of structural modes. However, more complex models that incorporate a larger number of harmonic terms, especially those with distinct and higher frequencies, introduce significant challenges. In such cases, the resulting input functions can induce system responses characterized by multiple high-frequency components, which may lead to instability and pronounced fluctuations in the system’s behavior. Moreover, the presence of multiple driving frequencies increases the risk of inducing resonance or near-resonance phenomena within the system.

On the other hand, a recent and effective approach is the use of polynomial input functions. These functions offer smoothness, continuity, and simplicity in satisfying both the initial and final conditions of the system. However, systems with multiple conditions often require higher-degree polynomials, making the precision of the polynomial constant coefficients critical, as they significantly influence the calculations. Additionally, higher-degree polynomials substantially increase computation time. Furthermore, for maneuvering times exceeding one second, substituting final time ( > 1 sec ) into high-degree polynomials can yield excessively large false magnitudes unless the polynomial constant coefficients are computed with high numerical accuracy. This issue also arises when computing earlier maneuvering times significantly less than one, where higher-degree polynomial terms disproportionately reduce these small time-values, resulting in inaccurate results. A further complication arises when analyzing the system response to such inputs in closed-form. The interaction between the system’s periodic behavior and the polynomial input generates integral terms of the form ∫ t n cos ω t d t , which become increasingly difficult to evaluate analytically as the polynomial degree n increases. Consequently, deriving a closed-form response or applying inverse Laplace transformations becomes extremely laborious and computationally intensive.

A suggested and promising input function is based on exponential terms. This type of function combines smoothness and continuity characteristic of polynomial inputs, while avoiding the complex integrals and intricate inverse Laplace transformations associated with higher-degree polynomials. For instance, terms of the form e − n t cos ω t are analytically tractable, allowing for straightforward integration and inversion in the Laplace domain, even for high values of n . As a result, exponential input functions offer a superior alternative to previously proposed methods, providing both mathematical simplicity and effective dynamic performance.

Robustness conception

Robust input shaping methods aim to mitigate the effects of system parameter uncertainties, particularly variations in cable length. One widely adopted approach, the Zero-Vibration-and-Derivative (ZVD) shaper, achieves robustness by flattening the sensitivity curve around a nominal cable length, ensuring a zero slope at the design point. This strategy is intended to slow the growth of residual oscillations as system parameters deviate from the nominal value. While effective in many scenarios, this approach faces two significant limitations.

First, a zero slope at the design point does not guarantee a gradual rise in the sensitivity curve. For instance, the function a x 2 has zero slope at x = 0 , yet it exhibits a steep increase for large values of a . Similarly, in ZVD shaping, certain parameter configurations can lead to abrupt increases in residual oscillations when the cable length extends slightly beyond the nominal range.

Second, although the ZVD shaper is designed to suppress residual oscillations near the nominal cable length, it does not necessarily minimize oscillations across the entire intended range. A more accurate metric for robustness is the integral of the sensitivity function over the range of interest lengths, namely the area under the sensitivity curve. This integral quantifies the total residual oscillation and provides a precise criterion for minimal sensitivity. ZVD and its extensions (ZVDD, ZVDDD, …) do not explicitly minimize this area and, therefore, may fall short in delivering optimal robustness.

In contrast, alternative shaping methods that directly minimize the area under the sensitivity curve, such as Minimum Vibration and Integral (MVI) and Zero Vibration and Minimum Integral (ZVI), offer improved efficiency and reliability in managing parameter uncertainties, especially in systems with variable cable lengths.

Through an advanced optimization algorithm, the ZVI shaper eliminates oscillation at the end of the maneuver for a specific cable length and significantly reduces oscillation amplitudes over a predefined nearby range by minimizing the area under the sensitivity curve. The MVI shaper, a special case of the ZVI shaper, does not require zero final conditions. Instead, it minimizes residual oscillations over a continuous specified range or at discrete cable lengths. ⁷

System’s initial conditions, energy, and power

The crane system’s initial conditions are crucial and informative. Typically, the trolley starts from rest, while the payload begins from a predefined angular position and velocity, often both set to zero. The system is then maneuvered to its final state, characterized by the trolley reaching a steady cruising speed and the payload achieving zero oscillations. Input shaping is applied during this transition period to suppress oscillations.

In the ideal case of a frictionless system, an assumption common in most research, minimizing the energy gained by the system, is generally infeasible during routine maneuvers. Once the initial and final conditions are defined, the total system’s energy is inherently fixed, leaving no room for energy optimality. A common misconception arises when the payload exhibits swinging and the trolley follows a time-varying acceleration profile, which may misleadingly suggest an opportunity for energy optimization. It is important to emphasize that this discussion pertains to energy, not power. Starting from typical initial conditions, zero velocity and no oscillation, and ending with the trolley at a constant cruising speed v c and the payload exhibiting zero oscillation implies a total energy input of 1 2 m v c 2 , where m is the combined mass of the trolley and payload. Therefore, minimizing the system’s total energy under these conditions is not feasible.

On the other hand, the power aspect is more debatable. The input power to the system primarily depends on the trolley’s velocity profile and the total maneuvering time. While the input power can be considered for optimization, shaping the input such that the trolley oscillates back and forth during the maneuver can inject unnecessary power into the system. Alternatively, the required power can theoretically be reduced by infinitely increasing the maneuver duration, this approach is impractical, as payload transportation fundamentally relies on time-efficient maneuvers.

Force and acceleration

Newton’s proportional relationship between the input force to the trolley and its acceleration implies that, in the absence of external disturbances, a crane system operating at constant trolley cruising speed with zero payload oscillations requires zero input force.

However, in certain applications involving faster maneuvers, a Time-Optimal Rigid-Body (TORB) shaper may be employed during the acceleration phase. In such cases, typically involving insensitive payloads or empty light containers, zero oscillations during the cruising stage are not required. As a result, the system is allowed to oscillate throughout the cruising phase, with input shaping applied during the deceleration phase to suppress oscillations by the end of the maneuver. In such applications, although the trolley moves at a constant speed during the cruising phase, the simple force–acceleration proportionality can be misleading. Due to the alternating magnitude and direction of the payload-cable tension, the motor must apply a variable and continuously adjusted force to maintain the trolley’s constant cruising speed.

Another interesting aspect is the motor power during the cruising phase. Since power is given by P = F v = m a v , the input power to the system is proportional to the product of velocity and acceleration. This implies nearly zero input power during constant cruising speed, where acceleration is zero. In ideal, frictionless conditions, this suggests that the motor can be shut down entirely during cruising phase ! However, some works inaccurately state that the trolley maintains full cruising speed with the motor operating at full capacity, overlooking the fact that, under constant speed and in the absence of disturbances, no continuous motor force or power are required.

A further observation worth noting is that, when assuming a massless trolley, the horizontal driving force applied by the motor is proportional to the horizontal acceleration of the swinging payload and is expressed as F = m ( u ¨ + L θ ˙ 2 sin θ − L θ ¨ cos θ ) . For small swing angles, this expression can be linearized to F = m ( u ¨ − L θ ¨ ) . Considering the simplified equation of motion L θ ¨ + g θ = u ¨ , the term L θ ¨ in the force expression can be replaced, yielding F = m g θ . This simplified expression provides a direct and convenient indication of the driving force acting on the system, revealing a positive proportionality between the force and the payload’s angular position.

Simplified input and single input-step

The shaped input to the system is a critical factor, valued for its effectiveness and speed. However, input simplicity also plays a significant role. In simple applications within small facilities, complex input functions requiring advanced control systems, actuators, and sensors are often undesirable and unaffordable. As a result, simplified input functions, despite being slower, are generally preferred.

Therefore, a compelling question arises: could a simple motor with primitive capacity support input shaping, possibly through a single, consistent input step, provided the system satisfies three final conditions: zero oscillation angle, zero angular velocity, and a specified trolley cruising speed? Considering a simplified crane model, a single-degree-of-freedom pendulum with zero initial conditions, applying a constant step input in trolley acceleration yields the response θ ( t ) = A ( 1 − cos ω t ) and θ ˙ ( t ) = A ω sin ω t , where A is a constant and ω is the system’s natural frequency.

The system’s response suggests that at the final time t f , when cos ω t f = 1 , implying sin ω t f = 0 , the payload oscillation is eliminated at the end of the maneuver. This condition can be easily achieved by adjusting the maneuvering time t f in coordination with the system’s natural frequency. Additionally, the constant A remains free to be tuned according to the desired trolley speed. Moreover, the system admits multiple valid solutions by selecting maneuver times as integer multiples of the oscillation period, thereby reducing the swing amplitude while extending the maneuver duration.

Yet, a two-step input function can also be implemented, offering a compromise between input simplicity and reduced maneuvering time. This approach provides effective oscillation suppression while allowing flexibility in selecting the maneuver duration, which can be arbitrarily defined. It also supports optimization of critical performance aspects such as jerk, maximum swing angle, and input power. Its structural simplicity makes it well-suited for complex models, including multi-pendulum systems, and it can account for actuator delays and control latencies, enhancing robustness in practical applications.

Final-swing dynamics

By the end of the maneuver, as the trolley approaches its final cruising speed and the payload nears its equilibrium angular position and velocity, namely zero, the interpretation of the direction conventions warrants careful examination. Hence, the linearized equation of motion for the system is given by L θ ¨ + g θ = u ¨ . Near the final time, θ approaches zero and becomes infinitesimally small, and thus u ¨ becomes approximately proportional to θ ¨ (Remark 1).

Now, consider the case where the payload swings from a negative angular displacement, implying a positive angular velocity. As this velocity approaches zero near equilibrium, the angular acceleration is negative. According to remark 1, this corresponds to a negative trolley acceleration. A negative trolley acceleration, while the trolley is approaching its final (positive) velocity, implies it is decelerating from a higher speed. However, in the common scenario where the trolley cruising velocity is the maximum velocity achieved during the motion, this deceleration contradicts the assumption. Therefore, such a situation, from a theoretical perspective, cannot occur physically.

On the other hand, in the opposite case, when the payload swings from a positive angular displacement toward the zero angle, the resulting angular velocity is negative but increasing toward zero, leading to positive angular acceleration near equilibrium. Consequently, the trolley acceleration, by remark 1, is also positive, indicating that the trolley is accelerating from a lower velocity toward its cruising velocity. This scenario aligns smoothly and logically with the natural dynamics of the system.

Therefore, at the closing stage of the maneuver, the payload is necessarily moving and swinging in the same direction as the trolley’s motion. It cannot be swinging against the trolley’s direction and simultaneously come to rest at θ = 0 without inducing oscillations.

Maneuvering time

Maneuvering time plays a critical role in system stability and the magnitude of input signals. Very short maneuvers require high-amplitude inputs, while reasonably longer maneuvers tend to relax the input magnitudes. However, excessively extended maneuver durations can lead to intolerable increases in input amplitude. This phenomenon becomes evident when the input function is generated analytically. As the maneuver time increases, the input amplitude initially decreases, reaching a minimum before rising again. Beyond a certain duration, this cycle repeats, input magnitudes decrease, reach a local minimum, and then increase once more.

From a mathematical perspective, the optimized input is obtained by solving a linear system of unknowns, typically step-input values, polynomial coefficients, or other input parameters. This system of equations comes from and represents the conditions that the system must satisfy.

For illustration, consider three conditions: final angular position, final angular velocity, and final trolley velocity. The setup matrices of these conditions yield a specific coefficient matrix. The determinant of the coefficient matrix in the system of unknowns plays a dominant role in determining the magnitude of the resulting input. At certain maneuver timings, a small determinant leads to large input amplitudes. These particular timings correspond to situations where multiple conditions are nearly linearly dependent, meaning that satisfying one condition almost guarantees satisfaction of another. This dependency is most prominent between the final angular position and angular velocity.

As the maneuvering time increases, there may exist a point at which the solution diverges, indicating mathematical redundancy among the condition equations. This redundancy shows as two nearly identical rows in the coefficient matrix. Physically, this implies that the shaper includes more input signals than necessary, that is, the system could achieve the desired final state with one fewer input, without compromising performance.

Input uniqueness

Shaped input often originates from mathematical formulations of the system’s equations of motion and its corresponding response. The conventional approach involves defining a general input function with undetermined constants, which are then determined or optimized based on the governing dynamics. However, a fundamental question arises: Is the resulting input unique for fixed system parameters and maneuvering time, assuming a fixed input structure, such as the number of steps, polynomial degree and number of terms, or a fixed driving frequency in the case of harmonic inputs? In fact, a stimulating result emerges in this context. Consider a simplified crane model with a shaped input of the form u ¨ ( t ) = f ( t ) , designed to eliminate residual oscillations at the end of the maneuver while satisfying the system’s final conditions. According to the convolution method, or Duhamel’s integral, the system response can be readily obtained by evaluating the integral θ ( t ) = 1 ω L ∫ 0 t f ( β ) sin ( ω ( t − β ) ) d β .

If the proposed shaped input f ( t ) eliminates oscillation by yielding zero from the integral, then a scaled input of the form α f ( t ) also eliminates oscillation, as the constant α does not affect the zero outcome of the integral. Therefore, a scaled shaped input remains valid but proportionally increases the trolley’s final speed. Hence, there are infinitely many scaled inputs that lead the system to its desired final state. This finding simplifies the mathematical formulation when adjusting the final trolley speed, as it merely requires scaling the existing input by an appropriate factor without the need to regenerate the entire shaper.

Extending this idea, if the input is expressed as a multiple-step function with unknown magnitudes to be determined or optimized, it can be normalized with respect to the first step, mathematically setting the amplitude of the first step to one. This normalization reduces the number of unknowns in the input formulation, simplifying the process of solving for zero oscillation at the end of the maneuver. Finally, the entire input sequence can be scaled by a correction factor to achieve the desired final trolley speed. This approach effectively reduces computational effort, complexity, and solution time.

Robust shapers

The concept of robust shapers is well-established in the field, particularly in addressing the sensitivity of oscillation amplitudes to uncertainties in parameters such as natural frequency. Achieving enhanced robustness often necessitates more sophisticated input functions, control strategies, and theoretical formulations. However, in the pursuit of simplicity, both in implementation and reduced maneuvering time, it may be reasonable to neglect the term of θ ˙ at intermediate stages. Doing so can significantly simplify the input structure, especially in the case of ZVD and higher-order robust shapers. This raises a critical question: is it worthwhile extending the maneuvering time, increasing input complexity, and possibly applying higher input amplitudes, merely for a marginal improvement in robustness by accounting for θ ˙ term?

By considering small oscillation swing amplitudes, the payload’s potential energy at its highest point at θ max is fully converted into kinetic energy at its lowest point at θ = 0 . This yields the relation θ ˙ 2 = 2 ω 2 ( 1 − cos θ max ) . Consequently, the expression for the square of the oscillation amplitude, typically written as θ 2 + θ ˙ 2 / ω 2 , becomes θ max 2 + 2 ( 1 − cos θ max ) . Expanding 2 ( 1 − cos θ max ) using a Taylor series yields θ max for small angles, resulting in a total amplitude expression of approximately 2 θ max with high accuracy. Therefore, implementing robust shapers with fewer added input terms, and accounting for robustness based solely on the swing angle, yields simplified yet reliable robust shapers. This approach also indirectly reduces the sensitivity to the influence of the payload’s angular velocity.

Another observation regarding robust shapers is that they effectively reduce swing amplitudes and exhibit lower sensitivity to variations in the system’s natural frequency, particularly due to variations in cable length. However, part of this reduced sensitivity appears to stem from the longer maneuvering times typically associated with robust shapers, compared to non-robust ones such as the ZV shaper. The extended maneuver duration tends to relax the system response, leading to smaller swing amplitudes and diminished sensitivity to cable length variations.

While the inherent design of robust shapers plays the primary role in enhancing robustness, the longer maneuvering time also contributes to this effect. Therefore, a fair comparison of robustness should involve a non-robust shaper (ZV) operated with an extended maneuvering time equivalent to that of a robust shaper (ZVD).

Double-pendulum model

A subtle but important detail arises when defining the input function for a double-pendulum model. It is instructive to begin with the single-pendulum model and then extend the analysis to the double-pendulum case.

In the simplified model’s equation of motion, L θ ¨ + g θ = u ¨ , as the system reaches its final state with zero angular position and velocity, and the trolley input acceleration terminates, the equation reduces to θ ¨ = 0 . This outcome reinforces the system’s stability and confirms the suppression of payload oscillations.

However, in the double-pendulum system, when proceeding similarly, the angular velocity and trolley input acceleration terms vanish from the equations of motion, resulting in the reduced form M [ θ ¨ 1 θ ¨ 2 ] T = 0 , M = [ ( m 1 + m 2 ) L 1 2 m 2 L 1 L 2 m 2 L 1 L 2 m 2 L 2 2 ] . In this case, if the first mass m 1 is assumed to be exceedingly small compared to the second mass m 2 , the determinant of the inertia matrix M becomes very small, potentially resulting in non-zero angular accelerations in practice, which implies instability or amplified sensitivity in the system’s response. Therefore, when implementing the double-pendulum model, this issue should be carefully considered. Actually, from a physical perspective, the system effectively behaves as a single pendulum, with the small first mass being negligible, resulting in fewer equations of motion, fewer final conditions to satisfy, and a reduced number of required input signals.

Output shaping

Output shaping is a revised form of input shaping that implements inverse dynamics procedures. Unlike input shaping, which assumes a predefined input function, output shaping specifies a desired output function representing the payload’s swing motion. The corresponding input function is then derived by substituting the output function into the system’s equations of motion. However, an essential consideration when assuming the output function, since it represents the nonhomogeneous solution of the equation of motion, is that it must be carefully designed to eliminate the homogeneous solution. Otherwise, the homogeneous response may emerge and degrade the effectiveness of output shaping. This issue becomes more complex when dealing with systems that have nonzero initial conditions.

Another important consideration is that the output function must yield zero angular acceleration at the final time of the maneuver when substituted into the equation of motion. Otherwise, any nonzero angular acceleration will excite the system and induce residual swing motion.

Nonzero initial conditions

Systems with nonzero initial conditions have recently been implemented to represent specific applications. Shapers that are robust to variations in initial conditions are therefore desirable. However, ZVD shapers designed with respect to initial conditions appear to be infeasible. From an energy perspective, specifying initial and final conditions imposes corresponding initial and final energy levels in the system. The difference between these energy levels represents the net work performed by the shaper, either added to or extracted from the system. Consequently, the tailored shaper delivers this precise amount of energy. Any deviation from the assumed initial conditions alters the system’s total energy, resulting in a mismatch that appears as residual kinetic energy, observed as oscillatory motion. Therefore, no shaper can provide robustness to variations in initial conditions, as the resulting residual oscillation is, from an energy standpoint, proportional to deviations in the initial conditions.

On the other hand, ZVD shapers designed for robustness to variations in cable length or natural frequency do not influence the system’s energy. This is because changes in cable length are unrelated to the system’s total energy, given that the initial and final conditions are predefined.

Conclusion

In conclusion, input shaping for overhead cranes involves a rich interaction of control strategies, modeling features, and mathematical aspects. Addressing subtle but significant factors is essential for insightful understanding of input shaping concepts, emphasizing the influence of system dynamics, and enabling practical, high-performance implementation.

Future research should expand input shaping techniques to a broader range of flexible systems beyond overhead cranes. Performance evaluation should also incorporate power consumption and actuator effort, motivating the development of comprehensive performance indices. Human factors require shaping methods that accommodate operator inputs and potential faults to enhance safety. Standardized benchmarks, reference models, and evaluation protocols are essential for systematic development and meaningful comparison.