Flexible Parallel Manipulator Research Articles

Intelligent controller for robust orientation control of smart actuator based parallel manipulator

Flexible parallel manipulators can be used as an orienting device due to its precise positioning ability. The flexibility demands the use of non-conventional actuators and noncontact type/flexible sensors for feedback measurement. This study proposes a novel 2 Degree of Freedom (DoF) parallel manipulator actuated by shape memory alloy springs (SMA) which could be used as an orienting device for a table top Cartesian robot. To mitigate the use of contact type sensor, model based feedback is proposed to obtain SMA length for the feedback control. The inclusion of SMA actuators to parallel mechanism imposes control challenges due to nonlinearities, coupling effects, and disturbances. This work also proposes a novel control scheme based on concept of partition control law to control the SMA spring position without the use of any complex mathematical model. The proposed controller is a combination of two parts Fuzzy PID servo control and the time delay estimation (TDE) based auxiliary control. The fuzzy PID control handles the error tracking and the auxiliary control tries to cancel the nonlinearities and disturbances, thus enhancing the robustness. The performance of the proposed controller is tested for a complex trajectory under disturbances and the results are compared with existing nonlinear controllers which proves superior nature of the proposed controller.

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.

Mechatronics Design and Kinematic Analysis of SMA Spring Actuated Parallel Manipulator

Flexible parallel manipulators can be used as an orienting device because of their precise positioning ability and better stiffness. The flexibility to the manipulator induces challenges in the selection of actuator, sensors and control. The paper presents a novel mechatronics design of a flexible symmetric parallel manipulator with minimum number of actuators used as an orienting device. Shape memory alloy springs (SMA) are used as an actuator because of ease of integration and higher deflection. In order to avoid any contact type sensor which could affect the compliance of the manipulator, the position of SMA spring is calculated indirectly from the kinematic model with the end effector orientation data measured from IMU sensor. The complete experimental set up design including electrical and control circuitry is explained in detail in the work. Preliminary open loop test is conducted on the prototype to understand the SMA response to various current input and to verify the effectiveness of the kinematic model in the actuator length measurement. Closed loop control based on ON/OFF control is implemented in the controller and is tested for step input.

Kinetostatics modeling and analysis of parallel continuum manipulators

Parallel continuum manipulators (PCMs) have increasingly attracted attentions in the field of mechanism and robotics, owing to their inherent merits, such as compliance in structure, reduction of backlash, and easiness in implementation. This paper presents a general approach for the kinetostatics modeling and analysis of spatial PCMs. A discretization-based technique is employed for the nonlinear large-deflection problems of the coupled slender flexible links in these flexible parallel manipulators. Benefiting from the mechanism approximation to the force-deflection characteristics of the flexible links, the kinetostatics models of the PCMs can be established analytically in the realm of robot kinematics/statics, rather than continuum mechanics. Moreover, a closed-form solution to the gradient of these nonlinear algebraic equations can be derived, such that gradient-based searching algorithms can be implemented to efficiently determine the equilibrium configurations in a variety of given conditions. A prototype of a three-limb 6-DOF PCM is developed, on which validation experiments are conducted. And the results verify the effectiveness of the proposed method for the kinetostatics modeling and analysis of parallel continuum manipulators.

Design and analysis of an active 2-DOF lockable joint

This article presents a new design of a 2-DOF active lockable joint, which can be applied in parallel reconfigurable manipulators as well as in other applications such as haptic devices. The direct and inverse kinematic problems of the proposed joint were considered in detail, and the kinematic features of the input cranks are presented. Velocity equations, Jacobians, and singular positions of the joint were thoroughly considered. The workspace of the joint was analyzed by utilizing performance indices. A quasi-static analysis was carried out, providing the output force distribution along the workspace. The finite element simulation was carried out in Ansys Workbench, in order to identify the most loaded parts of the joint and verify its strength. A prototype, manufactured using 3D printing, has proven the functionality of the concept. The prototype was used as a haptic device in order to control a five-bar flexible parallel manipulator, which was being developed by the CompMech research group.

Modeling and Simulation of a Shape Memory Alloy Spring Actuated Flexible Parallel Manipulator

Parallel platforms are known for its positioning ability. Adding flexibility to a robotic system gives an advantage to better interact with the environment. In this paper a 2 DOF flexible parallel platform has been developed which consist of a triangular top and base plate connected by a universal joint at its centroid. The Shape memory alloy (SMA) springs are used as an actuator because of its large strain and are connected between the top and base plate on all the three vertices of the platform. The top plate provides desired orientation with respect to base plate on SMA actuation. The work intends to understand the system behavior by developing a mathematical model of the system. Two separate models, one to understand the kinematics and dynamics of the platform and other to understand the dynamics of the SMA spring actuator has been developed. The models were then integrated to understand the complete system behavior. The paper focuses on the integration of various system level equations to develop a general model of the SMA actuated platform and the algorithm incorporated for the same has been discussed in the work. The mathematical models were developed in MATLAB/Simulink. The kinematic model of the platform was verified on the experimental set up. The dynamic model of SMA actuator described by Liang was used and the results obtained were found to characterize the SMA. The integrated model simulation results were also verified on the experimental set up.

Dynamic modeling and hierarchical compound control of a novel 2-DOF flexible parallel manipulator with multiple actuation modes

Abstract This paper addresses the problem of rigid-flexible coupling dynamic modeling and active control of a novel flexible parallel manipulator (PM) with multiple actuation modes. Firstly, based on the flexible multi-body dynamics theory, the rigid-flexible coupling dynamic model (RFDM) of system is developed by virtue of the augmented Lagrangian multipliers approach. For completeness, the mathematical models of permanent magnet synchronous motor (PMSM) and piezoelectric transducer (PZT) are further established and integrated with the RFDM of mechanical system to formulate the electromechanical coupling dynamic model (ECDM). To achieve the trajectory tracking and vibration suppression, a hierarchical compound control strategy is presented. Within this control strategy, the proportional-differential (PD) feedback controller is employed to realize the trajectory tracking of end-effector, while the strain and strain rate feedback (SSRF) controller is developed to restrain the vibration of the flexible links using PZT. Furthermore, the stability of the control algorithm is demonstrated based on the Lyapunov stability theory. Finally, two simulation case studies are performed to illustrate the effectiveness of the proposed approach. The results indicate that, under the redundant actuation mode, the hierarchical compound control strategy can guarantee the flexible PM achieves singularity-free motion and vibration attenuation within task workspace simultaneously. The systematic methodology proposed in this study can be conveniently extended for the dynamic modeling and efficient controller design of other flexible PMs, especially the emerging ones with multiple actuation modes.

Smooth adaptive sliding mode vibration control of a flexible parallel manipulator with multiple smart linkages in modal space

This paper addresses the dynamic model and active vibration control of a rigid-flexible parallel manipulator with three smart links actuated by three linear ultrasonic motors. To suppress the vibration of three flexible intermediate links under high speed and acceleration, multiple Lead Zirconium Titanate (PZT) sensors and actuators are collocated mounted on each link, forming a smart structure which can achieve self-sensing and self-actuating. The dynamic characteristics and equations of the flexible link incorporated with the PZT sensors and actuator are analyzed and formulated. The smooth adaptive sliding mode based active vibration control is proposed to suppress the vibration of the smart links, and the first and second modes of the three links are targeted to be suppressed in modal space to avoid the spillover phenomenon. Simulations and experiments are implemented to validate the effectiveness of the smart structures and the proposed control laws. Experimental results show that the vibration of the first mode around 92 Hz and the second mode around 240 Hz of the three smart links are reduced respectively by 64.98%, 59.47%, 62.28%, and 45.80%, 36.79%, 33.33%, which further verify the multi-mode vibration control ability of the smooth adaptive sliding mode control law.

Nonlinear State Estimation for Trajectory Tracking of a Flexible Parallel Manipulator

Lightweight robots can be advantageous when considering the lower energy consumption, the possibility to use smaller motors and the lower material cost. Nevertheless, in such lightweight structures non-negligible flexibilities are inherent which can lead to significant oscillations making the control more difficult. Within this research end-effector trajectory tracking of a parallel manipulator with a highly flexible link is considered. As centerpiece, a new approach to estimate the state of the system of differential algebraic equations is discussed to obtain the end-effector position and velocity. For the estimator, the kinematic loop is treated by a projection tangential to the constraint manifold which is based on a QR decomposition. Subsequently, an Unscented Kalman Filter is applied and the dependent coordinates are obtained by satisfying the algebraic constraint equations with the Newton-Raphson method. The utilized signals for the estimator are on the one hand position measurements of the actuators which are realized as direct drives and on the other hand measurements of strain gauges attached to the long and highly flexible link. As application, a basic end-effector output controller based on an equivalent rigid model and the transfer function of the dominant first bending eigenmode is utilized to enhance the tracking results. The concepts are applied to an experimental rig for validation purposes and to show first results for end-effector estimation with feedback control.

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 Model and Vibration Characteristics of Planar 3-RRR Parallel Manipulator with Flexible Intermediate Links considering Exact Boundary Conditions

Due to the complexity of the dynamic model of a planar 3-RRR flexible parallel manipulator (FPM), it is often difficult to achieve active vibration control algorithm based on the system dynamic model. To establish a simple and efficient dynamic model of the planar 3-RRR FPM to study its dynamic characteristics and build a controller conveniently, firstly, considering the effect of rigid-flexible coupling and the moment of inertia at the end of the flexible intermediate link, the modal function is determined with the pinned-free boundary condition. Then, considering the main vibration modes of the system, a high-efficiency coupling dynamic model is established on the basis of guaranteeing the model control accuracy. According to the model, the modal characteristics of the flexible intermediate link are analyzed and compared with the modal test results. The results show that the model can effectively reflect the main vibration modes of the planar 3-RRR FPM; in addition the model can be used to analyze the effects of inertial and coupling forces on the dynamics model and the drive torque of the drive motor. Because this model is of the less dynamic parameters, it is convenient to carry out the control program.

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.

A comparative study of elastic motions in trajectory tracking of flexible RPR planar manipulators moving with high speed

SUMMARYThe 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.

Optimal Point-to-Point Motion Planning of Flexible Parallel Manipulator with Multi-Interval Radau Pseudospectral Method

Optimal point-to-point motion planning of flexible parallel manipulator was investigated in this paper and the 3RRR parallel manipulator is taken as the object. First, an optimal point-to-point motion planning problem was constructed with the consideration of the rigid-flexible coupling dynamic model and actuator dynamics. Then, the multi-interval Legendre–Gauss–Radau (LGR) pseudospectral method was introduced to transform the optimal control problem into Nonlinear Programming (NLP) problem. At last, the simulation and experiment were carried out on the flexible parallel manipulator. Compared with the line motion of quantic polynomial planning, the proposed method could constrain the flexible displacement amplitude and suppress the residue vibration.

Hybrid force and motion control of flexible joint parallel manipulators using inverse dynamics approach

An inverse dynamics control algorithm is developed for hybrid motion and contact force trajectory tracking control of flexible joint parallel manipulators. First, an open-tree structure is considered by the disconnection of adequate number of unactuated joints. The loop closure constraint equations are then included. Elimination of the joint reaction forces and the other intermediate variables yield a fourth-order relation between the actuator torques and the end-effector position and contact force variables, showing that the control torques do not have an instantaneous effect on the end-effector contact forces and accelerations because of the flexibility. The proposed control law provides simultaneous and asymptotically stable control of the end-effector contact forces and the motion along the constraint surfaces by utilizing the feedback of positions and velocities of the actuated joints and rotors. A two degree of freedom planar parallel manipulator is considered as an example to illustrate the effectiveness of the method.

Vibration Suppression of Flexible Parallel Manipulator Based on Sliding Mode Control with Reaching Law

A discrete sliding mode control based on reaching law is investigated to suppress vibration of high-speed flexible manipulator. The finite element method and experimental modal test is applied to obtain the dynamic model of the system. Due to the influence from uncertain external disturbances and measurement noise, and uncertain parameters, sliding mode control has invariance of the sliding mode. The discrete sliding mode control approach with reaching law is applied to design the vibration controller which enables the system to be zero state as the system states are away from the equilibrium one due to external disturbances. The discrete Kalman filtering technique is employed to construct state estimator because the state variables cannot be directly measured. The first three natural frequencies and damping ratio are obtained by using experimental modal testing. The experimental control system is constructed and experimental validation is carried out using dSPACE real-time simulation system and MATLAB/Simulink. The experimental results showed that the proposed controller can effectively suppress the vibration response of the flexible manipulator.

Simulation and feed‐forward control of a flexible parallel manipulator

AbstractThe importance of lightweight constructions are steady increasing since they promise a low energy consumption together with higher movement speeds. However these demand modern, model‐based feed‐forward control designs. Especially the undesired vibrations due to the reduced overall stiffness of such manipulators have to be taken into account. A convenient way to model the dynamical behavior of systems that perform large, nonlinear motions superposed with small, elastic deformations is the floating frame of reference approach in a flexible multibody system. The application of the Newton‐Euler‐Formalism together with D'Alembert's principle to parallel manipulators results in a set of differential‐algebraic equations. Therefore, the consideration of the trajectory tracking problem with so‐called servo constraints appears to be promising. In case of a non‐flat system, the arising set of differential‐algebraic equations, which consists of the system dynamics, the holonomic loop closing constraint equations and the servo constraints embodies nontrivial dynamics. With an oblique projection, the embedded set of ordinary differential equations describing the internal dynamics can be obtained. The stability properties of these dynamics determines the complexity of the feed‐forward control design, as two‐point boundary value problems might have to be solved. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Dynamic Properties of Flexible Parallel Manipulators in the Presence of Type 2 Singularities

In the present paper, we expand information about the conditions for passing through Type 2 singular configurations of a parallel manipulator. It is shown that any parallel manipulator can cross the singular configurations via an optimal control permitting the favorable force distribution, i.e., the wrench applied on the end-effector by the legs and external efforts must be reciprocal to the twist along with the direction of the uncontrollable motion. The previous studies have proposed the optimal control conditions for the manipulators with rigid links and flexible actuated joints. The different polynomial laws have been obtained and validated for each examined case. The present study considers the conditions for passing through Type 2 singular configurations for the parallel manipulators with flexible links. By computing the inverse dynamic model of a general flexible parallel robot, the necessary conditions for passing through Type 2 singular configurations are deduced. The suggested approach is illustrated by a 5R parallel manipulator with flexible elements and joints. It is shown that a 16th order polynomial law is necessary for the optimal force generation. The obtained results are validated by numerical simulations carried out using the software ADAMS.

Hybrid Multi-Mode Input Shaping Suppresses the Vibration of a 3-DOF Parallel Manipulator

The methodologies and application of hybrid multi-mode positive impulses input shaping of a 3-DOF flexible parallel manipulator in this paper. First, the structural system and the dynamic equations are expressed for a 3-DOF manipulator. Second, the hybrid multi-mode positive impulses input shaping is introduced to reduce the residual vibration of the multi-mode system or to decrease the time-delay of the system response at the same time. The theory of the hybrid multi-mode positive impulses input shapers are presented, and the hybrid two-mode positive impulses input shapers of a 3-DOF manipulator are established and compared with the classic multi-mode positive impulses input shapers. Finally, the numerical simulations are made, and the robust of the input shapers are presented and compared.

Hybrid position/force control of a flexible parallel manipulator

In this paper, simultaneous position/force control of a closed-chain planar manipulator with the last link flexible is studied when the manipulator is in contact with an environment. The proposed manipulator consists of a flexible link connected to three rigid linkages whichare optimized for kinematic and force manipulability in the region of interest. The flexible link is modeled as a series of rigid links connected by virtual torsion springs. A hybrid position/force control algorithm is developed and implemented on the manipulator. Experimental results are presented to verify the performance of the controller.