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This paper proposes a novel control scheme for precise path tracking control in cable driven parallel robots (CDPRs) with axially-flexible cables, with particular focus to the challenging case of cable suspended parallel robots (CSPRs). To handle model nonlinearities while ensuring small computational effort, a controller made by two sequential control actions is developed. The first term is a position-dependent, model predictive control (MPC) with embedded integrator to compute the optimal cable tensions ensuring accurate path tracking and fulfilling the feasibility constraints; bounds on the feasible tensions are also included. The second control term transforms the optimal tensions into the commanded motor torques, and hence currents, that are evaluated through the kinetostatic model of the electric motors used for winding and unwinding the cables. Control design is performed through the robot dynamics model, formulated with the assumption of rigid cables. Moreover, the proposed control strategy is presented in two different architectures, collocated control and noncollocated control. Flexibility is handled by penalizing large tension variations in the cost function adopted in the controller design, plus some hard constraints on the maximum tension derivatives. These features, together with the embedding of the integrator within the MPC formulation, ensure smooth control tensions that allow handling the axial flexibility of the cables, although it is not explicitly considered in the controller design.To assess the performances of the proposed control algorithm, a kinematically-determined robot with a suspended, lumped end-effector is simulated by also adopting very flexible cables. Additionally, a simplified dynamic model of the electrical dynamics and the sensor quantization are included to provide a realistic representation of the real environments. The results, together with the fair comparison with a benchmark, corroborate the effectiveness of the proposed approach, its robustness, and its feasibility in real-time controllers due to the wise reduction of the computational effort.

Purpose This paper proposes an integrated method for optimizing the response of underactuated linear vibratory feeders operating in open-loop control, under generic periodic excitations. The goal is ensuring a uniform motion of the tray, despite the presence of less actuators than degrees of freedom and of several specifications of the desired motion. Method To cope with the underactuated nature of these systems and with their non-minimum phase behavior, dynamic structural modification and the inverse dynamics approach are properly integrated by exploiting a common definition of the system internal dynamics. In the inverse dynamics problem, the inverse dynamics is stabilized through output redefinition and the resulting ordinary differential equations are integrated to compute causal actuation forces, ensuring almost-exact tracking for as many coordinates as the number of actuators. The tracking of the remaining coordinates of interest is improved through a proper design of the mechanical parameters, based on the modification of the internal dynamics. Results The effectiveness of the proposed method is assessed through numerical simulations performed on the challenging case of a 14-degrees of freedom underactuated non-minimum phase linear vibratory feeder adopted in manufacturing plants to convey products. Conclusion The results evidence the benefits obtained by integrating structural modification together with inverse dynamics. Inverse dynamics is effective, since the tracking error on the imposed coordinates is negligible. On the other hand, the benefits introduced by dynamic structural modification are proved as well, by the reduction of the tracking error also for the non-imposed coordinates.

An enhanced inverse dynamics approach is here presented for feedforward control of underactuated multibody systems, such as mechanisms or robots where the number of independent actuators is smaller than the number of degrees of freedom. The method exploits the concept of partitioning the independent coordinates into actuated and unactuated ones (through a QR-decomposition) and of linearly combined output, to obtain the internal dynamics of the nonminimum-phase system and then to stabilize it through proper output redefinition. Then, the exact algebraic model of the actuated sub-system is inverted, leading to the desired control forces with just minor approximations and no need for pre-actuation. The effectiveness of the proposed approach is assessed by three numerical test cases, by comparing it with some meaningful benchmarks taken from the literature. Finally, experimental verification through an underactuated robotic arm with two degrees of freedom is performed.

Conventional model-based design methods are often limited in their effectiveness by model-plant discrepancy. A solution to this problem is proposed in this work to enhance the robustness of motion planning solution for systems affected by parametric uncertainty. The method exploits a variational formulation in the form of a two-point boundary value problem (TPBVP) in which the robustness is achieved as a constraint enforced at the two boundaries. The formulation, which is specifically targeted at underactuated systems, aims at reducing both the transient and residual vibrations, as well as at mitigating the actuation effort. The development of the method is supported by its application to two numerical test-cases in the form of a double pendulum on a cart and a translating flexible beam.

Purpose This paper proposes a motion planning technique for precise path and trajectory tracking in an underactuated, non-minimum phase, spatial overhead crane. Besides having a number of independent actuators that is smaller than the number of degrees of freedom, tip control on this system presents unstable internal dynamics that leads to divergent solution of the inverse dynamic problem. Method The paper exploits the representation of the controlled output as a separable function of the actuated (i.e., the platform translations) and unactuated (i.e., the swing angles) coordinates to easily formulate the internal dynamics, without any approximation, and to study its stability. Then, output redefinition is adopted within the internal dynamics to stabilize it, leading to stable and causal reference commands for the platform translations. Results Besides proposing the theoretical formulation of this novel method, the paper includes the numerical validation and the experimental application on a laboratory setup. Comparison with the state-of-the-art input shaping is also proposed. Conclusion The results, obtained through different reference trajectories, clearly show that almost exact tracking is obtained also in the experiments, by outperforming the benchmarks.

Vibration absorption is a core research area in the design and control of structures and machines, and exploiting antiresonances is an effective approach for systems under harmonic excitation. This paper proposes a comparative study and a review of the main passive techniques to antiresonance assignment proposed in the recent literature, by discussing them through some numerical examples too. The techniques discussed include the well-known Tuned Mass Damper, which has been widely developed in the literature. However, as the title reveals, great attention is paid to the methods inherited from the field of dynamic structural modification that assign antiresonances without modifying the number of degrees of freedom, by exploiting a proper modification of the system inertial and stiffness parameters. Due to higher mathematical complexity, these approaches have been less investigated in the literature although they are an effective and less invasive approach to antiresonance assignment, especially for machines. To puzzle out the complicated subject matter of antiresonances, their background and their key features are also discussed by reviewing the main theoretical results and their relationship with the assignment techniques. The paper is also enriched with several numerical examples to compare different methods and investigate the features of antiresonances. The concluding remarks of the paper bring together some open issues in this field of research and outlines some possible research directions.

The paper proposes a multi-domain approach to the optimization of the dynamic response of an underactuated vibrating linear system through eigenstructure assignment, by exploiting the concurrent design of the mechanical properties, the regulator and state observers. The approach relies on handling simultaneously mechanical design and controller synthesis in order to enlarge the set of the achievable performances. The underlying novel idea is that structural properties of controlled mechanical systems should be designed considering the presence of the controller through a concurrent approach: this can considerably improve the optimization possibilities. The method is, first, developed theoretically. Starting from the definition of the set of feasible system responses, defined through the feasible mode shapes, an original formulation of the optimality criterion is proposed to properly shape the allowable subspace through the optimal modification of the design variables. A proper choice of the modifications of the elastic and inertial parameters, indeed, changes the space of the allowable eigenvectors that can be achieved through active control and allows obtaining the desired performances. The problem is then solved through a rank-minimization with constraints on the design variables: a convex optimization problem is formulated through the “semidefinite embedding lemma” and the “trace heuristics”. Finally, experimental validation is provided through the assignment of a mode shape and of the related eigenfrequency to a cantilever beam controlled by a piezoelectric actuator, in order to obtain a region of the beam with negligible oscillations and the other one with large oscillations. The results prove the effectiveness of the proposed approach that outperforms active control and mechanical design when used alone.

In this paper a novel method to improve the residual vibration suppression in underactuated uncertain flexible systems through motion planning is proposed. The proposed technique relies on the concurrent use of input shaping and on the alteration of the mechanical properties of the system, which enables the enhancement of the robustness to uncertain parameters. Robustness is formulated through the parametric sensitivities of the natural frequencies that provide an analytical and non-probabilistic tool, which is embedded in the eigenstructure assignment algorithm developed in this work. The effectiveness of the proposed method is assessed through a benchmark testbed composed by a triple pendulum attached to a delta robot, that should execute rest-to-rest, residual vibration-free motion. Hence, residual vibrations of the pendulum are used to measure the accuracy and the sensitivity of the proposed combined shaper-IDSM approach. Both the numerical and experimental results confirm the advantages of the proposed technique for the suppression of the residual vibrations in uncertain systems over traditional techniques. In particular, robustness of the dominant tuning frequency is increased with respect to variations of the tip-end mass of the pendulum, representing a varying payload transferred by a robotic system.

Antiresonance assignment is an effective approach to vibration absorption in lightweight systems under harmonic excitation. The goal is shifting an antiresonance frequency of a receptance to match the one of the excitation force, for making a system coordinate experiencing no (or very small) forced vibrations. A novel method for antiresonance assignment through active control is proposed and validated in this paper. The method relies on the unit-rank output feedback control technique to shift one antiresonance. The assignment problem is solved for both point and cross-receptances exploiting either collocated or non-collocated control architectures. The gain computation is done analytically, either through the system mass, stiffness and damping matrices, or through the measured receptances. Two method extensions are also proposed. First, the output feedback technique is extended to account for an additional sensor, to perform the simultaneous pole-zero assignment based on the zero non-spillover condition. Second, to allow for larger frequency shifts without reducing too much the stability margins, a passive-active, hybrid strategy is proposed. In the first stage, modification of the inertial and elastic parameters is employed to shift the antiresonance frequency as more as possible; in the second stage, the prescribed antiresonance is exactly assigned by means of active control, with smaller gains compared to active control alone. Experimental validation of the method is proposed through a smart, flexible cantilever beam controlled by a non-collocated pair made by an on-the-shelf piezoelectric actuator and a laser vibrometer.

This paper proposes a method for active vibration control to a two-link flexible robot arm in the presence of time delay, by means of robust pole placement. The issue is of practical and theoretical interest as time delay in vibration control can cause instability if not properly taken into account in the controller design. The controller design is performed through the receptance method to exactly assign a pair of pole and to achieve a given stability margin for ensuring robustness to uncertainty. The desired stability margin is achieved by solving an optimization problem based on the Nyquist stability criterion. The method is applied on a laboratory testbed that mimic a typical flexible robotic system employed for pick-and-place applications. The linearization assumption about an equilibrium configuration leads to the identification of the local receptances, holding for infinitesimal displacements about it, and hence applying the proposed control design technique. Nonlinear terms, due to the finite displacements, uncertainty, disturbances, and the coarse encoder quantization, are effectively handled by embedding the robustness requirement into the design. The experimental results, and the consistence with the numerical expectations, demonstrate the method effectiveness and ease of application.

This paper describes a method for planning energy-efficient trajectories for industrial robots driven by brushless or DC motors with regenerative braking. The optimization problem is defined upon spline interpolation methods, using piecewise polynomial functions to produce a trajectory passing through a sequence of via-points, and on the electromechanical model of the robot. The formulation introduced in this work is aimed at estimating and optimizing the energy consumption using closed-form expressions and therefore without the need for any numerical integration of the robot dynamics. The method accounts for kinematic constraints on speed, acceleration, and jerk, as well as constraints due to the limitations of the power supply and of the regenerated energy storage system.

Vibration absorption is a core research topic in structural dynamics and the mechanics of machines, and antiresonance assignment is an effective solution to such a problem in the presence of harmonic excitation forces. Due to recent developments in the theory of feedback control systems, the use of active control approaches to antiresonance assignment has been recently gaining more attention in the literature. Therefore, several methods exploiting state feedback or output feedback have been proposed in recent years. These techniques that just rely on servo-controlled actuators are becoming an interesting alternative to active approaches that emulate the well-known Tuned Mass Damper in an active (or semi-active) framework. This paper reviews and compares the most important approaches, with a greater focus on the methods exploiting the concept of control theory without adding new degrees of freedom in the system. The theoretical results, with the underlying theory, are discussed to highlight the key features of each assignment techniques. Several numerical examples where different techniques are applied and compared, also providing some analysis usually neglected in the literature, enrich the paper and demonstrate the key concepts.

Resonant vibration generators, such as vibratory feeders or ultrasonic sonotrodes, are often employed in manufacturing to generate harmonic vibrations with suitable amplitude, spatial shape, and frequency, in order to meet the process requirements. These underactuated systems are usually excited in open loop by few actuators, and therefore, it is not ensured that the desired response is correctly achieved, since the feasible motions should belong to the subset of the allowable motions. To achieve the closest approximation of the desired vibrations, some new solutions are here proposed. The first strategy is the optimal shaping of the harmonic forces exerted by the actuators, by solving an inverse dynamic problem through a coordinate transformation and the projection of the desired response onto the subspace of the allowable motion. By exploiting the formulation of such a subspace, a second approach that involves concurrently both the force shaping and the modification of the inertial and elastic system parameters is proposed. The idea of this approach is to exploit the modification of the elastic and inertial parameters to properly shape the allowable subspace in such a way that it spans the desired response. A solution method is developed, and analytical sensitivity analysis is proposed to choose the design variables. Validation is proposed through a linear vibratory feeder with a long flexible tray, taken from the literature. The results show the effectiveness of the proposed strategies that lead to a very precise approximation of the desired response.

In this paper a method for planning smooth and energy-efficient trajectories for industrial robots is presented. The motion design, which is based on the use of piecewise polynomial functions, is optimized for achieving minimum energy consumption when executing a trajectory that passes through a sequence of via-points with \(C^3\) continuity. For robots with simple kinematics, such as the Cartesian robot presented in this work, the energy consumption estimation can be performed using equations based on inverse dynamic models that allow fast and reliable numerical computing.

This paper proposes a novel adaptive model-based filtering technique for dynamic mass measurement through load cells, named Adaptive Shaper-Based Filter (ASBF). Filtering is performed by convolving the oscillating load cell signal with a baseline of few impulses (usually 3) to rapidly compensate for its underdamped behaviour. The correct timings and amplitudes of the impulses are computed by means of a simplified dynamic model of the load cell (that requires the knowledge of the natural frequency and the damping ratio of the empty cell), and the filter output itself (i.e. the estimated mass). The load cell is modelled through the theory of systems with variable mass as a linear time-variant system and the variations of frequency and damping are predicted; in this way, the filter zeros track the load cell poles to cancel them. Robustness specifications are included in the filter design to account for the unavoidable uncertainty and estimation errors. Given the non-rational transfer function of the proposed Adaptive Shaper Based Filter, whose poles have an infinite and negative real part, rapid settling time is ensured. Experimental assessment is proposed by comparing the results with some benchmark filters.

This work proposes to exploit functional redundancy as a tool to enhance the energy efficiency of a robotic system. In a functionally redundant system, i.e., one in which the number of degrees of freedom required to complete the task is smaller than the number of available degrees of freedom, the motion of the extra degrees of freedom can be tailored to enhance a performance metric. This work showcases a method that can be used to effectively enhance the energy efficiency through motion design, using a detailed dynamic model of the UR5 serial robot arm. The method is based on an optimization of the motion profile, using a parametrized description of the end-effector orientation: the results showcase an increased efficiency that allows energy savings up to 20.8%, according to the energy consumption results according to the electro-mechanical dynamic model of the robot.

This paper proposes an inverse structural modification method for the assignment of antiresonances in undamped vibrating systems by modifying the inertial and elastic properties of the existing degrees of freedom of the original system. Hence, no additional degrees of freedom are added to the system. The problem is formulated as an eigenstructure assignment approach since such a novel formulation is suitable for complex systems, such as those modeled through finite elements. Indeed, these systems are difficult to handle with the methods already proposed in the literature. Additionally, the proposed approach is suitable for both point and cross receptances. Assignment is cast as a constrained non-convex non-linear minimization problem and the proposed solving strategy is based on the homotopy optimization approach. The method effectiveness is shown through three meaningful test cases.

The ever-growing interest toward energy efficiency imposes the optimization of mechanism design under an energetic point of view. Even if the benefit of using spring balancing systems to reduce energy consumption is intuitive, the relation between spring design and electrical energy consumption has never been systematically addressed in the literature, which is mainly focused on static compensation of gravity forces. This paper tackles this novel and important issue and proposes an analytical method for model-based design of springs minimizing the energy required in rest-to-rest motion. The method relies on the model of energy dissipation that accounts for the characteristics of the mechanical, electrical, and power electronic components of a servo-actuated mechanism. The theory is developed with reference to a single rotating beam. The proposed solution ensures significant energy saving compared with the traditional static balancing design of springs and is particularly suitable for repetitive (cyclic) motion tasks.

This paper proposes a novel method for pole placement in linear vibrating systems through state feedback and rank-one control. Rather than assigning all the poles to the desired locations of the complex plane, the proposed method exactly assigns just the dominant poles, while the remaining ones are free to assume arbitrary positions within a pre-specified region in the complex plane. Therefore, the method can be referred to as &ldquo, regional pole placement&rdquo, A two-stage approach is proposed to accomplish both the tasks. In the first stage, the subset of dominant poles is assigned to exact locations by exploiting the receptance method, formulated for either symmetric or asymmetric systems. Then, in the second stage, a first-order model formulated with a reduced state, together with the theory of Linear Matrix Inequalities, are exploited to cluster the subset of the unassigned poles into some stable regions of the complex plane while keeping unchanged the poles assigned in the first stage. The additional degrees of freedom in the choice of the gains, i.e., the non-uniqueness of the solution, is exploited through a semidefinite programming problem to reduce the control gains. The method is validated by means of four meaningful and challenging test-cases, also borrowed from the literature. The results are also compared with those of classic partial pole placement, to show the benefits and the effectiveness of the proposed approach.

Eigenvalues and eigenvectors play a fundamental role in determining the dynamic behavior of a vibrating system. Thus, an important goal in vibration control is to modify the eigenstructure to match the design specifications. Feedback control is a popular approach to this problem, which is however not always satisfactory. In fact, while system controllability suffices to assure that any requirement on the eigenvalues can be met by the controlled system, desired closed-loop eigenvectors cannot be attained in general, due to inherent limitations of active control. To overcome such a problem, this paper proposes a hybrid approach in which the mechanical system and the controller are concurrently designed to improve the attainability of the desired eigenstructure. Indeed, the suitable modification of the system inertial and elastic parameters modifies the set of eigenvectors that can be achieved through active control. In this work is demonstrated that such an objective can be effectively accomplished by minimizing the rank of a certain matrix which depends on the features of the original system and on the desired eigenpairs. Two algorithms for rank minimization are described and adjusted for the problem under consideration. The method is validated with three examples. In the first one, eigenstructure assignment of a 5 degrees of freedom system that previously appeared in the literature is performed, demonstrating the effectiveness of the proposed approach with respect to the state of the art. In the second example the same 5 degrees of freedom system is considered by also including damping, to evaluate the method capability to deal with damped systems. The third one is a 30 degrees of freedom system that enables the comparison of the two algorithms for different choices of the actuation.