The study of inertial forces effects at high speeds in flexible parallel manipulators, which generate undesired deviations, is a challenging task due to the coupled and complicated equations of motion. A dynamic model of the Revolute Prismatic Revolute (RPR) planar manipulators (specifically 3-RPR, 2-RPR and 1-RPR) with flexible intermediate links is developed based on the assumed mode method. The flexible intermediate links are modeled as Euler-Bernoulli beams with fixed-free boundary conditions. Using the Lagrange multipliers, a generalized set of differential algebraic equations (DAEs) of motion is developed. In the simulations, the rigid body motion of the end-effector is constrained by some moving constraint equations while the vibrations of the flexible intermediate links cause deviations from the desired trajectory. From this analysis, the dynamic performance of the manipulators when tracking a desired trajectory is evaluated. A comparison of the results indicates that in some cases, adding each extra RPR chain in the n-RPR planar manipulators with flexible intermediate links reduces the stiffness and accuracy due to the inertial forces of the flexible links, which is opposite to what would be expected. The study provides insights to the design, control and suitable selection of the flexible manipulators.

In this paper, a closed-form dynamic model of flexible manipulators is developed, based on the Newton–Euler formulation of motion equations of flexible links and on the adoption of the spatial vector notation. The proposed model accounts for two main innovations with respect to the state of the art: it is obtained in closed form with respect to the joints and modal coordinates (including the quadratic velocity terms) and motion equations of the whole manipulator can be computed for any arbitrary shape of the links and any possible link cardinality starting from the output of several commercial (finite element analysis) FEA codes. The Newton–Euler formulation of motion equations in terms of the joint and elastic variables greatly improves the simulation performances and makes the model suitable for real-time control and active vibration damping. The model has been compared with literature benchmarks obtained by the classical multibody approach and further validated by comparison with experiments collected on an experimental manipulator.

This paper presents the dynamic modeling of a 3-RPR planar parallel manipulator with three flexible intermediate links in order to investigate the effects of the intermediate links flexibility on the undesired vibrations of the end-effector. For this purpose, the intermediate links are modeled as Euler--Bernoulli beams with two types of fixed-pinned and fixed-free boundary conditions based on the assumed mode method (AMM). The equations of motion of the 3-RPR manipulator are formulated using the augmented Lagrange multipliers method in the form of differential algebraic equations (DAEs) by incorporating the elastic and rigid coordinates in the set of generalized coordinates. After defining the initial conditions and imposing external forces to the manipulator, the equations are then solved numerically using the Modified Extended Backward-Differentiation Formula Implicit (MEBDFI) approach. Comparison of the simulation results for two different boundary conditions shows clearly the effects of flexibility of the intermediate links on the vibration of the end-effector trajectory. Results of this work can be used for the dynamic modeling of other manipulators or to design a controller for reducing the undesired vibrations.

In this study, modelling and control of a two-link robot manipulator whose first link is rigid and the second one is flexible is considered for both land and underwater conditions. Governing equations of the systems are derived from Hamilton's Principle and differential eigenvalue problem. A computer program is developed to solve non-linear ordinary differential equations defining the system dynamics by using Runge–Kutta algorithm. The response of the system is evaluated and compared by applying classical control methods; proportional control and proportional + derivative (PD) control and an intelligent technique; integral augmented fuzzy control method. Modelling of drag torques applied to the manipulators moving horizontally under the water is presented. The study confirmed the success of the proposed integral augmented fuzzy control laws as well as classical control methods to drive flexible robots in a wide range of working envelope without overshoot compared to the classical controls.

The effect of axial forces on the dynamic properties is formulated and investigated for a 3-PRR planar parallel manipulator with three flexible intermediate links. A dynamic model of the manipulator system is developed based on the assumed mode method with the consideration of the effect of longitudinal forces on lateral stiffness is included. The flexible intermediate links are modeled as Euler–Bernoulli beams with pinned-pinned boundary conditions, which are verified by experimental modal tests. Natural frequencies of bending vibration of the intermediate links are derived as the functions of axial force and rigid-body motion of the manipulator. Dynamic behavior including the effect of axial forces on lateral deformation is investigated, and configuration-dependant frequencies are analyzed. Numerical simulations of configuration-dependent frequency properties and axial forces are performed to illustrate the effect of axial forces on the dynamic behaviors of the flexible parallel manipulator. Simulation results of mode amplitudes, deformations, axial forces, inertial, and coupling forces are presented, and further validate the theoretical derivations. These analyses and results provide a new and valuable insight to the design and control of the parallel manipulators with flexible intermediate links.

A dynamic model for two three-link cooperating structurally-flexible robotic manipulators is presented in this study. The equations of motion are derived using the extended Hamilton's principle and Galerkin's method, and must satisfy certain geometric constraints due to the closed chain formed by the two manipulators and the object. The dynamic model presented here is for the purpose of designing controllers. Therefore, a low-order model which captures all the major effects is of interest. Computer simulated results are presented for the case of moving an object along an elliptical path using the two cooperating flexible manipulators.

This paper addresses the dynamic simulation and control of structural vibrations of a 3-PRR parallel manipulator with three flexible intermediate links, to which are bonded lead zirconate titanate (PZT) actuators and sensors. Flexible intermediate links are modelled as Euler–Bernoulli beams with pinned-pinned boundary conditions. A PZT actuator controller is designed based on strain rate feedback (SRF) control. Control moments from PZT actuators are transformed to force vectors in modal space and are incorporated in the dynamic model of the manipulator. The dynamic equations are developed based on the assumed mode method for the flexible parallel manipulator with multiple PZT actuator and sensor patches. Numerical simulation is performed and the results indicate that the proposed active vibration control strategy is effective. Spectral analyses of structural vibrations further illustrate that deformations from structural vibration of flexible links are suppressed to a significant extent when the proposed vibration control strategy is employed, while the deflections caused by inertial and coupling forces are not reduced.

Dynamic simulation of a flexible manipulator performing physical contact (including low-speed impact) tasks with stiff environment is very time consuming because very small integration step sizes have to be used for numerical stability. Existing model order reduction techniques cannot be readily applied due to the nonlinear nature of the contact dynamics. In this paper, a method is introduced to deal with this problem. The method first linearizes the contact force model on the right-hand side of the dynamics equations periodically. It then identifies the linear “stiffness” and “damping” terms from the linearized contact force model and combines them with the existing structural stiffness and damping matrices of the associated multibody system on the left-hand side of the equations. After such a process, the traditional modal analysis and reduction techniques for linear dynamic systems can be applied to reduce the order of the resulting dynamic system. Two numerical examples of flexible manipulators performing a contact task are presented to demonstrate the significant gain in computational efficiency and the improved output results.

A survey of the field of control for flexible multi-link robots is presented. This research area has drawn great attention during the last two decades, and seems to be somewhat less “attractive” now, due to the many satisfactory results already obtained, but also because of the complex nature of the remaining open problems. Thus it seems that the time has come to try to deliver a sort of “state of the art” on this subject, although an exhaustive one is out of scope here, because of the great amount of publications. Instead, we survey the most salient progresses – in our opinion – approximately during the last decade, that are representative of the essential different ideas in the field. We proceed along with the exposition of material coming from about 119 included references. We do not pretend to deeply present each of the methods quoted hereafter; however, our goal is to briefly introduce most of the existing methods and to refer the interested reader to more detailed presentations for each scheme. To begin with, a now well-established classification of the flexible arms control goals is given. It is followed by a presentation of different control strategies, indicating in each case whether the approach deals with the one-link case, which can be successfully treated via linear models, or with the multi-link case which necessitates nonlinear, more complex, models. Some possible issues for future research are given in conclusion.

This paper focuses on overcoming the problem of tracking control in structurally flexible redundant manipulators by utilizing their self-motion capabilities. In the proposed algorithm, the self-motion is evaluated in order to nullify the dominant modal force of flexural motion that is induced by a rigid body motion.

The flexure motions of manipulators, which are induced by joint motion, cause undesired inaccuracy in end-effector tracking. In-plath planning states, joint trajectories are so designed as not to excite but to damp out the flexure motions. The self-motion, inherent in redundant manipulators, can alter joint motion, influencing the flexure motion (by exciting and damping the flexure modes), while not affecting end-effector motion at all. Therefore, the self-motion can be utilized to regulate flexibility and effectively reduce the end-effector tracking error.

The effectiveness and applicability of the proposed algorithm have been demonstrated through numerical simulation with three-link planar robotic manipulators possessing flexible links.