Planar Flexible Manipulator Research Articles

Study on modeling and dynamic performance of a planar flexible parallel manipulator based on finite element method.

The application of a high-speed parallel manipulator necessitates the adoption of a lightweight design to reduce dead weight. However, this increases the elastic deformation of certain components, affecting the dynamic performance of the system. This study examined a 2-DOF planar flexible parallel manipulator. A dynamic model of the parallel manipulator composed of fully flexible links was established using a floating reference coordinate system and a combination of the finite element and augmented Lagrange multiplier methods. A dynamic analysis of the simplified model under three driving torque modes showed that the axial deformation was less than the transverse deformation by three orders of magnitude. Further, the kinematic and dynamic performance of the redundant drive was significantly better than that of the non-redundant drive, and the vibration was well suppressed in the redundant drive mode. In addition, the comprehensive performance of driving Mode 2 was better than that of the other two modes. Finally, the validity of the dynamic model was verified by modeling via Adams. The modular modeling method is conducive to the extension to other models and programming. Furthermore, the dynamic model of the established fully flexible link system can aid in optimizing the lightweight design and dynamic performance of the parallel manipulator.

Dynamic formulation of a two link flexible manipulator and its comparison analysis with a knuckle joint cantilever

In the present study a dynamic modelling technique for a two-link flexible planar manipulator is presented. The developed manipulator model can include an arbitrary number of flexible links. The gross equation of motion of each flexible link has been obtained by composing the rigid body motion and a small elastic deformation of each link. In order to bring the simplicity the modelling technique has been applied to model a two link flexible manipulator. The analytical derivation and final expression for the equation of motion has been shown for a two link flexible manipulator using the proposed modelling method. Later on, a comparison analysis has been studied between the proposed manipulator model and a simple cantilever with a knuckle joint. Finite element analysis of the beam has been done in ANSYS 17.1 considering different types of elements under different loading conditions. The various numerical results have been generated at different nodal points by taking the origin of the Cartesian coordinate system at the fixed end of the beam. Finally, a comparison analysis has been made to find a way to validate the proposed mathematical modelling.

Modeling and Vibration Reduction of a Flexible Planar Manipulator with Experimental System Identification

Modeling and Vibration Reduction of a Flexible Planar Manipulator with Experimental System Identification

Trajectory planning of a spatial flexible manipulator for vibration suppression

Vibration decreases operational accuracy and productivity of flexible manipulators, and trajectory planning is an effective way to suppress vibration. However, the existing trajectory planning methods can only suppress vibration of planar flexible manipulators. In this paper, a trajectory planning method is proposed to suppress vibration of spatial flexible manipulators. Firstly, the dynamic models of each link of the flexible manipulator are established separately. Then with the constraint equations, the dynamic model of the flexible manipulator is established as a Differential Algebraic Equation (DAE). Secondly, the trajectory functions are designed as quintic polynomials, and the conditions are deduced to satisfy acceleration limits of each joint. Finally, the trajectory planning problem is transferred to an optimal problem. Particle Swam Optimization (PSO) is adopted to solve the optimal problem. Numerical simulation is conducted to demonstrate the good performance of the proposed method.

A stable reentry trajectory for flexible manipulators

Trajectory tracking of fast-lightweight manipulators is an emerging topic. These systems often possess an unstable internal dynamics that is related to the inverse dynamics problem. The main contributions of this work are (i) a direct relation between internal dynamics variables and trajectory constraints as state equations in an observer canonical form; (ii) the entry/reentry trajectory defined as input control in order to stabilise the unstable internal dynamics; (iii) the methodology is well-suited for systems modelled in a finite element approach, related to an index-3 differential algebraic equation; (iv) a linearised design is established, but the results are suitable to consider a nonlinear model predictive control of underactuated/flexible manipulators, since a reentry trajectory is often required to minimise the output error recursively. For the application, (a) a planar underactuated manipulator with one passive joint and (b) a planar flexible manipulator on a cart were considered.

Dynamic Modeling and Development of Symbolic Calculation Software for N‐DOF Flexible‐Link Manipulators Incorporating Lumped Mass

Considering the complex dynamic modeling of multi‐DOF planar flexible manipulators, a general‐purpose method for the rigid‐flexible coupling dynamic modeling of N‐DOF flexible manipulators is proposed in this paper, and symbolic calculation software is developed. The modeling method is based on the Lagrange equation and assumed mode method (AMM). First, the N‐DOF flexible manipulator is divided into two parts, which are assumed to be rigid and flexible. On this basis, the rigid part and the flexible part are coupled, and the calculation process of the model is further simplified. Then, the simplest general symbolic expression of the dynamic model of the N‐DOF flexible manipulator is obtained with the induction method. According to the modeling method, “symbolic expression computation software for dynamic equations of N‐DOF flexible manipulators” is developed using the symbolic calculation software Mathematica. Finally, the dynamic modeling method and the symbolic calculation software are verified by a trajectory tracking experiment with a PD control applied to a 2‐DOF flexible manipulator. Compared with the traditional modeling method, the calculation time can be reduced by more than 90% using the modeling method proposed in this paper, which significantly reduces the complexity of the modeling process.

Adaptive neural network output feedback control for flexible multi-link robotic manipulators

ABSTRACTIn this paper, a novel approach for adaptive control of flexible multi-link robots in the joint space is presented. The approach is valid for a class of highly uncertain systems with arbitrary but bounded dimension. The problem of trajectory tracking is solved through developing a stable inversion for robot dynamics using only joint angles measurement; then a linear dynamic compensator is utilised to stabilise the tracking error for the nominal system. Furthermore, a high gain observer is designed to provide an estimate for error dynamics. A linear in parameter neural network based adaptive signal is used to approximate and eliminate the effect of uncertainties due to link flexibilities and vibration modes on tracking performance, where the adaptation rule for the neural network weights is derived based on Lyapunov function. The stability and the ultimate boundedness of the error signals and closed-loop system is demonstrated through the Lyapunov stability theory. Computer simulations of the proposed robust controller are carried to validate on a two-link flexible planar manipulator.

Rigid-flexible coupling dynamic modeling and investigation of a redundantly actuated parallel manipulator with multiple actuation modes

A systematic dynamic modeling methodology is presented to develop the rigid-flexible coupling dynamic model (RFDM) of an emerging flexible parallel manipulator with multiple actuation modes. By virtue of assumed mode method, the general dynamic model of an arbitrary flexible body with any number of lumped parameters is derived in an explicit closed form, which possesses the modular characteristic. Then the completely dynamic model of system is formulated based on the flexible multi-body dynamics (FMD) theory and the augmented Lagrangian multipliers method. An approach of combining the Udwadia-Kalaba formulation with the hybrid TR-BDF2 numerical algorithm is proposed to address the nonlinear RFDM. Two simulation cases are performed to investigate the dynamic performance of the manipulator with different actuation modes. The results indicate that the redundant actuation modes can effectively attenuate vibration and guarantee higher dynamic performance compared to the traditional non-redundant actuation modes. Finally, a virtual prototype model is developed to demonstrate the validity of the presented RFDM. The systematic methodology proposed in this study can be conveniently extended for the dynamic modeling and controller design of other planar flexible parallel manipulators, especially the emerging ones with multiple actuation modes.

Dynamic Analysis of Flexible-Link Planar Parallel Manipulator with Platform Rigidity Considerations

This paper presents dynamic analysis studies of planar parallel flexible 3-RRR manipulator with and without considering the flexibility of mobile platform. Initially, by treating all the members of the manipulator as flexible, the joint displacements, reaction forces and stresses are obtained during a specified trajectory tracking in Cartesian space. A comparative study is conducted with manipulator configuration having rigid mobile platform using coupled dynamics of limbs and kinematic constraints of mobile platform. Dynamic response of flexible manipulator is validated using ANSYS simulations for two different cases of trajectories. The results show a remarkable effect of flexibility of mobile platform on the overall dynamic response. After validation of the model, the inverse dynamic analysis data is used to create the system dynamics by employing generalized regression neural network (GRNN) model and the forward dynamic solutions of the flexible manipulator are predicted instantaneously. This study is useful for the real time implementation of motion control of flexible manipulators with complex dynamic model of manipulators.

Dynamic Performance Evaluation of Serial and Parallel RPR Manipulators with Flexible Intermediate Links

This paper presents a comprehensive study of the workspace, dynamic characteristics and accuracy of three planar flexible manipulators with 3-RPR, 2-RPR and 1-RPR structures moving at high speed. A geometrical procedure is employed to obtain the workspaces of the manipulators. The flexible intermediate links are modeled as the Euler–Bernoulli beams with fixed-free boundary conditions based on the assumed mode method. Using the Lagrange multipliers, a generalized set of differential algebraic equations of motion is developed for the planar RPR manipulators. Three moving constraints, which are obtained from an inverse kinematics analysis and applied to the actuated base joints, impose the end-effector to follow a high-speed circular motion as the desired trajectory. From this analysis, the dynamic performance of 1-RPR flexible serial manipulator and 2-RPR and 3-RPR flexible parallel manipulators in tracking a desired trajectory is evaluated. Based on the results, it is concluded that, in addition to the specific structure of the manipulator, the accuracy generally depends on the operation conditions. The results contest the general assertion which claims that parallel manipulators have more accuracy and stiffness than serial counterparts.

Tracking Accuracy Analysis of a Planar Flexible Manipulator With Lubricated Joint and Interval Uncertainty

A method for trajectory tracking accuracy analysis of a two-link flexible manipulator with lubricated revolute joint involving interval uncertainty is presented. In this method, first, fuzzy self-tuning proportion integration differentiation (PID) control is applied to track the desired tip trajectory of the manipulator. The absolute nodal coordinate formulation (ANCF) is employed for the finite element discretization of the flexible manipulator. And lubricated revolute joint is modeled based on the infinitely short journal bearing with Gümbel conditions. Second, uncertainty of clearance size is considered, and interval analysis method is applied. Numerical simulation is posted to investigate the cushioning effect of lubricants on the clearance and influence of uncertainty on control accuracy of the manipulator. The results show that the lubricants can improve the stability of motion and operation precision of the manipulator; however, uncertainty of the manipulator may reduce the control accuracy of the manipulator.

Chaotic Vibration Prediction of a Free-Floating Flexible Redundant Space Manipulator

The dynamic model of a planar free-floating flexible redundant space manipulator with three joints is derived by the assumed modes method, Lagrange principle, and momentum conservation. According to minimal joint torque’s optimization (MJTO), the state equations of the dynamic model for the free-floating redundant space manipulator are described. The PD control using the tracking position error and velocity error in the manipulator is introduced. Then, the chaotic dynamic behavior of the manipulator is analyzed by chaotic numerical methods, in which time series, phase plane portrait, Poincaré map, and Lyapunov exponents are used to analyze the chaotic behavior of the manipulator. Under certain conditions for the joint torque optimization and initial values, chaotic vibration motion of the space manipulator can be observed. The chaotic time series prediction scheme for the space manipulator is presented based on the theory of phase space reconstruction under Takens’ embedding theorem. The trajectories of phase space can be reconstructed in embedding space, which are equivalent to the original space manipulator in dynamics. The one-step prediction model for the chaotic time series and the chaotic vibration was established by using support vector regression (SVR) prediction model with RBF kernel function. It has been proved that the SVR prediction model has a good performance of prediction. The experimental results show the effectiveness of the presented method.

Modeling and Control of Flexible Telescoping Manipulators

This paper considers modeling and control of a planar flexible two-link telescoping manipulator. The telescoping interface between the deploying and non-deploying portions of the links is discussed in detail. The equations of motion are derived using Lagrange's equation and constrained using a projection method that eliminates the need to compute Lagrange multipliers explicitly. Passivity-based control of the telescoping manipulator is also investigated. It is shown that several passive input-output maps exist. In particular, by a suitable redefinition of the inputs and outputs, a modified tip-based control is made possible. The model is validated through numerical simulation, and the tip-based control is compared with joint-based control. Simulation results show that tip-based control has better closed-loop performance compared with joint-based control.

On the control of a single flexible arm robot via Youla-Kucera parameterization

SUMMARYIn this paper, based on the Youla-Kucera (Y-K) parameterization, the control of a flexible beam acting as a flexible robotic manipulator is investigated. The method of Youla parameterization is the simple solution and proper method for describing the collection of all controllers that stabilize the closed-loop system. This collection comprises function of the Youla parameter which can be any proper transfer function that is stable. The main challenge in this approach is to obtain a Youla parameter with infinite dimension. This parameter is approximated by a subspace with finite dimensions, which makes the problem tractable. It is required to be generated from a finite number of bases within that space and the considered system can be approximated by an expansion of the orthonormal bases such as FIR, Laguerre, Kautz and generalized bases. To calculate the coefficients for each basis, it is necessary to define the problem in the form of an optimization problem that is solved by optimization techniques. The Linear Quadratic Regulator (LQR) optimization tool is employed in order to optimize the controller gains. The main aim in controller design is to merge the closed-loop system and the second order system with the desirable time response characteristic. The results of the Youla stabilizing controller for a planar flexible manipulator with lumped tip mass indicate that the proposed method is very efficient and robust for the time-continuous instances.

Position Control Strategy and Experimental Research of A 2R Underactuated Planar Flexible Manipulator

A simple position control strategy in stages is presented for the underactuated planar manipulator with passive joints and flexible links. The assumed mode dynamic model is established of a 2R underactuated manipulator with a second flexible link and a second passive joint. The dynamic coupling characteristics and the controllability of this system are analyzed. The position control strategy of this manipulator is investigated in stages based on the switch of the brake at the passive joint. And the corresponding algorithms of both the active and passive joints are designed by using the PID method. Finally, on the basis of a self-designed experimental setup of an underactuated planar manipulator, the position control of this 2R underactuated planar flexible manipulator is experimentally completed. The simulative and experimental results illustrate that the active and passive joints can track the targets perfectly in respective stage and the operational task of this manipulator can be implemented, which verify the effectiveness of the proposed method and the correctness of the dynamic model of the underactuated flexible manipulator.

Dynamic Modeling of Planar Flexible Multi-Link Manipulators with Accounting for both Link Foreshortening and Link Material Damping

In order to investigate dynamic performance of flexible multi-link manipulators more exactly, establishing the dynamic model with accounting for link foreshortening and link material damping is needed. In this paper, a new dynamic model for planar flexible multi-link manipulators is established by using Lagrange approach. Both link foreshortening and link material damping are accounted for in this model. As a case simulation, this model is applied to a planar flexible two-link manipulator with a tip mass, and the motion responses of the manipulator are obtained using Gear method.

Research and Simulation of Inverse Dynamics of a Wheeled Mobile Single Link Flexible Manipulator

The inverse dynamics modeling of a wheeled mobile single link flexible planar manipulator(WMSLFPM) is presented. Based on Bernoulli-Euler beam theory and under pin-free boundary conditions,the single link flexible manipulator is discretized. The system inverse dynamic model of the WMSLFPM,which is derived by using finite elements by virtue of Hermit interpolation function and Newton-Ouler equations and virtual principle,is concisely described in terms of matrices and vectors measured with respect to a body frame of reference. The numeric simulations are carried out to the inverse dynamic model of a WSMSLFPM in the frequency domain to get the driving torque/force and elastic configuration variables. This technique is suitable for the study on the inverse dynamics of mobile flexible manipulators under the conditions of high precision at the tip of light flexible arms and for building up both open-loop control and closed-loop control strategies.

Self-calibration Method of Planar Flexible 3-RRR Parallel Manipulator

Planar flexible parallel manipulators, which concentrate the merits of both flexible manipulators and lower-mobility parallel manipulators, have become a hot spot of current researches. A simple and effective self-calibration approach for a planar flexible 3-RRR parallel manipulator is proposed and verified. Based on the error modeling, parameter identification equations are derived by vector chains. Actual poses of the manipulator during applications are measured on line by linear wire encoders, which have been calibrated by instruments through standard positioning holes of the static platform. According to the identification results deduced from the theoretical trajectory of numerical control system, the control model is compensated. As a result, the accuracy of the manipulator is greatly improved, and the study of the self-calibration by linear wire encoders is completed. In view of the great availability of measuring implements and modeling method, as well as the capacity of measuring the actual poses on line, this method is a practical solution to self-calibration of lower-mobility planar flexible parallel manipulators and an available measure for the closed-loop control.

Passivity of a Two-Link Planar Flexible Manipulator Using an Exact Model Derived from a Recursive Dynamic Formulation

This paper provides a proof of passivity of a two-link planar flexible manipulator whose dynamics is described by a set of nonlinear ordinary and partial differential equations. We first propose a recursive dynamic formulation for multi-link planar flexible manipulators so that we have an exact model having simple structure of equation, which can simplify the proof of passivity in comparison with an exact model having complex structure derived by such as Hamilton's principle. We then perform passivity analysis of a two-link planar flexible manipulator by differentiating an energy-based storage function along with the exact model under the assumption that no external force or moment acts on the end-effector of the manipulator. The analysis results in that the flexible manipulator defines a passive map from the joint torques to the joint angles with respect to the energy-based storage function.

A Constrained Lagrange Formulation of Multilink Planar Flexible Manipulator

In this article, the closed-form dynamic equations of planar flexible link manipulators (FLMs), with revolute joints and constant cross sections, are derived combining Lagrange’s equations and the assumed mode shape method. To overcome the lengthy and complicated derivative calculation of the Lagrangian function of a FLM, these computations are done only once for a single flexible link manipulator with a moving base (SFLMB). Employing the Lagrange multipliers and the dynamic equations of the SFLMB, the equations of motion of the FLM are derived in terms of the dependent generalized coordinates. To obtain the closed-form dynamic equations of the FLM in terms of the independent generalized coordinates, the natural orthogonal complement of the Jacobian constraint matrix, which is associated with the velocity constraints in the linear homogeneous form, is used. To verify the proposed closed-form dynamic model, the simulation results obtained from the model were compared with the results of the full nonlinear finite element analysis. These comparisons showed sound agreement. One of the main advantages of this approach is that the derived dynamic model can be used for the model based end-effector control and the vibration suppression of planar FLMs.