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.

A negative input shaped command is presented for flexible systems to reduce the residual oscillation under unequal acceleration and braking delays of actuators that are common issues in industrial applications. Against this nonlinearity, a compensated unit magnitude zero vibration (UMZV) shaper is analytically developed with a phasor vector diagram and a ramp-step function to approximate the dynamic response of the unequal acceleration and braking delays of actuators. A closed-form solution is presented with a benchmark system without sacrificing the generality and simplicity for industrial applications. The robustness and control performance of the exact solution are numerically evaluated and compared with those of an existing negative input shaper in terms of the switch-on time, command interference, and effects of the shaper parameters. The proposed negative input shaped commands are experimentally validated with a mini-bridge crane.

This paper presents a simple analysis evaluating the stability threshold for magnetically levitated flexible structures using dissipative colocated controllers. It is shown that with such a control structure, the controller that stabilizes a rigid levitated mass can also stabilize a simple flexible structure with the same overall mass and electrodynamics. Experimental and simulation results are presented to validate this conclusion.

Input shaping accomplishes vibration reduction by slightly increasing the acceleration and deceleration periods of the command. The increase in the deceleration period can lead to system overshoot. This paper presents a new class of reduced-overtravel input shapers that are designed to reduce shaper-induced overtravel from human-operator commands. During the development of these new shapers, an expression for shaper-induced overtravel is introduced. This expression is used as an additional constraint in the input-shaper design process to generate the reduced-overtravel shapers. Experiments from a portable bridge crane verify the theoretical predictions of improved performance. Results from a study of eight industrial bridge crane operators indicate that utilizing the new reduced-overtravel input shapers dramatically reduces task completion time, while also improving positioning accuracy.

An efficient and computationally robust method for synthesis of component dynamics is developed. The method defines the interface forces/moments as feasible vectors in transformed coordinates to ensure that connectivity requirements of the combined structure are met. The synthesized system is then defined in a transformed set of feasible coordinates. The simplicity of form is exploited to effectively deal with modeling parametric and nonparametric uncertainties at the substructure level. Uncertainty models of reasonable size and complexity are synthesized for the combined structure from those in the substructure models. In particular, we address frequency and damping uncertainties at the component level. The approach first considers the robustness of synthesized flexible systems. It is then extended to deal with nonsynthesized dynamic models with component-level uncertainties by projecting uncertainties to the system level. A numerical example is given to demonstrate the feasibility of the proposed approach.

Input shaping is an effective means of eliminating vibration in many types of flexible systems. This paper discusses how input shaper performance is affected by a fixed acceleration limit. This type of limit is a common occurrence in many mechanical drive systems because it corresponds to a constant force or torque input. It is shown that some input shapers are not affected by an acceleration limit under certain conditions. A test criterion is developed to determine what types of input shapers are negatively affected, and a method is proposed to compensate for the detrimental effects of the constant acceleration limit. Experimental results from an industrial crane support the main theoretical results.

A technique for driving a flexible system with on-off actuators is presented and experimentally verified. The control system is designed to move the rigid body of a structure a desired distance without causing residual vibration in the flexible modes. The on-off control actions are described by closed-form functions of the system’s natural frequency, damping ratio, actuator force-to-mass ratio, and the desired move distance. Given the closed-form equations, the control sequence can be determined in real time without the need for numerical optimization. Performance measures of the proposed controller such as speed of response, actuator effort, peak transient deflection, and robustness to modeling errors are examined. Experiments performed on a flexible satellite testbed verify the utility of the proposed method.

Accounting for friction is important when designing controllers for precision motion control systems. However, the presence of the friction and the flexibility in the system yields undesirable behaviors such as residual vibration and stick-slip oscillation near the reference value. In the proposed development, a pulse amplitude modulated controller with user-specified pulse width, is used to initiate the motion so as to permit the system to coast to the desired final position after the final pulse, with zero residual vibrations. The proposed technique is illustrated on the floating oscillator benchmark problem, where friction acts on the first mass. Numerical simulation illustrates the effectiveness of the proposed technique.

Pulse width control refers to the use of a control law to determine the duration of fixed-height force pulses for point-to-point position control of a plant that is subject to mechanical friction, including stiction. A quantitative measure of the performance of a pulse width control system is introduced. Applications of this measure suggest that piecewise-linear-gain pulse width control laws will often provide better performance than constant-gain pulse width control laws. A method for designing piecewise-linear-gain pulse width control laws is introduced. The performance measure and piecewise-linear-gain control law design method are demonstrated in applications to the control of the position of the end-effector of an industrial robot.

This paper investigates a new design technique of input shaping filters for multi-input flexible systems using convex optimization synthesis techniques for finite impulse response filters (FIR filters). The objective of the input shaping filter design is to find the minimum length and the minimum number of nonzero impulses of the FIR filter that forces the system to track the reference command without any residual vibration, while satisfying additional performance and control constraints. This multi-objective optimization is solved using a two-step algorithm that sequentially solves two quasi-convex optimization problems. Compared with previously published nonlinear optimization approaches, this new approach does not require a priori knowledge of the forms of input shaping filters and enables much greater flexibility for including additional performance and robustness objectives. Furthermore, this convex-based approach can be applied to multi-input systems. The multiple input shaping filter has been experimentally verified on the Stanford University Two-Link Flexible Manipulator.

A method for generating on-off command profiles for flexible systems is presented. The command profiles move a system without residual vibration while using a specified amount of actuator fuel. Robustness to modeling errors can be incorporated into the design of the command signals. Techniques are presented that facilitate implementation and indicate prudent choices for the amount of fuel to be used. The method is compared to other command generation techniques that balance fuel usage and slew time.

Simple proofs are presented which show that hybrid boundary control systems are asymptotically, but not exponentially, stable. The asymptotic rates of decay of eigenvalues are determined analytically for both second and fourth-order systems.

The problem of scheduling strictly positive real (SPR) dynamic compensation for control of nonlinear flexible systems which exhibit collocated inputs and outputs is explored. The major application is the robust motion control of structurally flexible systems whose dynamics possess significant configuration dependence. Included in this class are flexible robot manipulators. The issue of designing a linear time-invariant SPR compensator for control of a nonlinear system is examined Controller performance is enhanced by scheduling a series of such designs and a scheduling algorithm is developed which preserves robust stability with respect to the nonlinear plant model Global asymptotic stability of equilibrium setpoints is proven when the scheduled SPR compensator is used in conjunction with a proportional feedback gain. A numerical example employing a two-link flexible manipulator is used to illustrate the approach and compare the efficacy of different scheduling algorithms.

Controller design for a rigid-flexible two-link manipulator is considered. Robustness of independent joint PD control is investigated. It has been shown that the stability of independent joint PD control does not depend explicitly on the system parameters. No discretization or linearization of the equations of motion is required to assure the stability. Simulation studies also show that independent joint PD control gives reasonably good results for the flexible system, and is robust to parameter uncertainties.

This paper describes a method for limiting vibration in flexible systems that have more than one characteristic frequency and mode. It is only necessary to have knowledge of the component mode frequencies and damping ratios in order to be able to calculate the timing and magnitudes of the impulse sequence used in the shaping. Only two impulses, in the nonrobust case, or three impulses in a more robust case, are necessary regardless of the number of component frequencies. Simple tests are established to determine when this technique can be used and examples are presented.

In this paper we derive formulas for computing the closed-loop time response of an uncertain flexible mechanical system with a truncated model. It is assumed that the dynamics of the flexible mechanical system is described by an infinite series of lightly damped vibration modes. The modal parameters, damping ratio, and natural frequency, are assumed to lie within known ranges. Given a truncated model and a robust control solution, our formulaes define upper and lower bounds for the time response of single-input/output systems about the closed-loop response of the truncated model.

Recent research studies noncolocated control of flexible mechanical systems using time delay. The developments are limited to undamped flexible systems; damped flexible systems have not been considered. This paper investigates noncolocated vibration control of a viscously damped string using time delay. The control system is formulated in the Laplace transform domain. Based on the understanding of the system eigenstructure, a modified Bode plot of the feedback controller is introduced in a design region. The Bode plot designed, along with a specific time delay in the feedback loop, proper sensor and actuator positions, and proper control gain, guarantees stabilization of vibration of the damped string.

In this paper we investigate generic properties of structural modeling pertinent to structural control, with emphasis on noncollocated systems. Analysis is performed on a representative example of a pinned-free Euler-Bernoulli beam with distributed sensors. Analysis in the wave number plane highlights the crucial qualitative characteristics common to all structural systems. High sensitivity of the transfer function zeros to errors in model parameters and sensor locations is demonstrated. The existence of finite right half plane zeros in noncollocated systems, along with this high sensitivity, further complicates noncollocated controls design. A numerical method for accurate computation of the transfer function zeros is proposed. Wiener-Hopf factorization is used to compute equivalent delay time, which is important in controls design.

The lumping approximation used frequently for dynamic analysis of distributed parameter systems is facilitated for a class of flexible systems by a technique using 4 × 4 coordinate transformation matrices to account for the deflection of elastic elements under load. This approach is used to develop the linear equations of spatial motion for a system of two rigid masses connected by a chain with an arbitrary number of massless beams and controlled joint rotations.