Compliant Parallel Manipulator Research Articles

A Single-Fidelity Surrogate Modeling Method Based on Nonlinearity Integrated Multi-Fidelity Surrogate

Abstract Surrogate model provides a promising way to reasonably approximate complex underlying relationships between system parameters. However, the expensive modeling cost, especially in large problem sizes, hinders its applications in practical problems. To overcome this issue, with the advantages of the multi-fidelity surrogate (MFS) model, this paper proposes a single-fidelity surrogate model with a hierarchical structure, named nonlinearity integrated correlation mapping surrogate (NI-CMS) model. The NI-CMS model first establishes the low-fidelity model to capture the underlying landscape of the true function, and then, based on the idea of MFS model, the established low-fidelity model is corrected by minimizing the mean square error to ensure prediction accuracy. Especially, a novel MFS model (named NI-MFS), is constructed to enhance the stability of the proposed NI-CMS model. More specifically, a nonlinear scaling term, which assumes the linear combination of the projected low-fidelity predictions in a high-dimensional space can reach the high-fidelity level, is introduced to assist the traditional scaling term. The performances of the proposed model are evaluated through a series of numerical test functions. In addition, a surrogate-based digital twin of an XY compliant parallel manipulator is used to validate the practical performance of the proposed model. The results show that compared with the existing models, the NI-CMS model provides a higher performance under the condition of a small sample set, illustrating the promising potential of this surrogate modeling technique.

Design and testing of a large-workspace XY compliant manipulator based on triple-stage parallelogram flexure

This paper proposes a new XY compliant parallel manipulator with large workspace and low coupling motion by resorting to planar arrangement of multistage parallelogram mechanisms. The triple-stage parallelogram flexure is chosen as the basic mechanism for motion transmission and decoupling. Its fixed and moving ends adopt planar and spatial connections, respectively. By this connecting scheme, more types of general multistage parallelogram flexures can be constructed. The driving stiffness and resonant frequency of the manipulator are derived based on statics analysis of theoretical mechanics and dynamics analysis via Lagrangian equation. The accuracy of the analytical model is confirmed by performing finite element analysis simulation, and the performance variances of three manipulators are compared. Moreover, a prototype of the XY compliant manipulator has been fabricated and tested. Results show that the workspace of the parallel manipulator is 1.51 cm × 1.48 cm, and the coupling ratios of the two axes are 1.11% and 1.06%, respectively. The volume ratio is introduced as a compactness indicator for comparison study versus existing designs. The centimeter-level stroke can promote the application of XY parallel manipulators in more industrial scenarios.

A Compact Mirror-Symmetrical XY Compliant Parallel Manipulator for Minimizing Parasitic Rotations

Abstract The design of XY compliant parallel manipulators (CPMs) remains challenging considering the tradeoff between mirror-symmetry for better constrained undesired rotations and a small footprint, although a significant number of XY CPMs have been reported in extensive applications. This paper presents a new XY CPM using mirror-symmetry without increasing its footprint, mainly aiming to reduce the undesired parasitic rotations of input and output motion stages. The concept of mirror-symmetry is deployed to tackle the parasitic rotations, with the help of using a multilayer compact XY CPM design method. A nonlinear and analytical model of the proposed XY CPM is derived using free body diagrams and the beam constraint model (BCM) to accurately analyze its performance characteristics over a large range of motion. The designed XY CPM is then verified by the nonlinear finite element analysis (FEA) method. Finally, the proposed multilayer design is fabricated using two pieces of aluminum plate and it is mounted in a measurement system to experimentally validate several performance characteristics. The analytical, FEA, and/or experimental results show that the proposed design can sufficiently constrain undesired motions including parasitic rotation of input and output stages, cross-axis coupling error, and actuator isolation index. Compared with exiting designs, the proposed design also shows its merits in large motion range and out-of-plane stiffness.

Mechatronic Model of a Compliant 3PRS Parallel Manipulator

Compliant mechanisms are widely used for instrumentation and measuring devices for their precision and high bandwidth. In this paper, the mechatronic model of a compliant 3PRS parallel manipulator is developed, integrating the inverse and direct kinematics, the inverse dynamic problem of the manipulator and the dynamics of the actuators and the control. The kinematic problem is solved, assuming a pseudo-rigid model for the deflection in the compliant revolute and spherical joints. The inverse dynamic problem is solved, using the Principle of Energy Equivalence. The mechatronic model allows the prediction of the bandwidth of the manipulator motion in the 3 degrees of freedom for a given control and set of actuators, helping in the design of the optimum solution. A prototype is built and validated, comparing experimental signals with the ones from the model.

Design of Four-DoF Compliant Parallel Manipulators Considering Maximum Kinematic Decoupling for Fast Steering Mirrors

Laser beams can fluctuate in four directions, which requires active compensation by a fast steering mirror (FSM) motion system. This paper deals with the design of four-degrees-of-freedom (DoF) compliant parallel manipulators, for responding to the requirements of the FSM. In order to simplify high-precision control in parallel manipulators, maximum kinematic decoupling is always desired. A constraint map method is used to propose the four required DoF with the consideration of maximum kinematic decoupling. A specific compliant mechanism is presented based on the constraint map, and its kinematics is estimated analytically. Finite element analysis demonstrates the desired qualitative motion and provides some initial quantitative analysis. A normalization-based compliance matrix is finally derived to verify and demonstrate the mobility of the system clearly. In a case study, the results of normalization-based compliance matrix modelling show that the diagonal entries corresponding to the four DoF directions are about 10 times larger than those corresponding to the two-constraint directions, validating the desired mobility.

Optimal design, modeling and control of a long stroke 3-PRR compliant parallel manipulator with variable thickness flexure pivots

Abstract The variable thickness flexure pivot (VTFP) is a promising flexure hinge to construct long stroke compliant mechanisms with high precision, since it combines both the advantages of the classical flexure pivot and the notched flexure hinge. In this paper, a 3-PRR compliant parallel manipulator (CPM) is proposed by employing the VTFPs to serve as the passive rotational joints. Geometric parameters of the VTFP and the 3-PRR manipulator are optimized by genetic algorithm to obtain the desired motion performance of the CPM. A prototype 3-PRR CPM is fabricated using the optimized results. In order to consider the parasitic rotational center shift of the VTFPs, an accurate inverse kinematic model (AIKM) is established, numerical results show the superiority of the AIKM compared with the rigid inverse kinematic model (RIKM). Moreover, an on-line learning radical basis function neural network (RBFNN) is established to compensate the unmodeled factors of the system, and a disturbance observer (DOB) is designed by utilizing the proposed AIKM and the RBFNN compensator. The observed external disturbances of the system is compensated via a feed-forward compensation strategy. Experiments show that the 3-PRR CPM can achieve micron scale trajectory tracking accuracy over centimeter's motion range and micro-degree rotational tracking accuracy by the proposed control scheme.

Optimal Design and Tracking Control of a Superelastic Flexure Hinge Based 3-PRR Compliant Parallel Manipulator

A major obstacle that restricts the application of notched flexure hinge based compliant mechanism is its limited motion range. This paper presents the optimal design and tracking control of a superelastic flexure hinge based 3-PRR compliant parallel manipulator (CPM) to achieve high precision planar motion within centimeter's translation range and up to 10 degrees' rotational range. Firstly, a novel asymmetric ellipse-parabola (AEP) notch shape is proposed, and the geometric parameters of the AEP superelastic flexure hinge are acquired via multi-objective optimization to obtain desirable transmission performance. Secondly, a nominal inverse kinematic model of the CPM is established, and the dimension parameters of the 3-PRR manipulator are synthesized to maximize the dexterity of the CPM over the regular workspace. Thereafter, a disturbance observer based inverse kinematic control scheme (DOB-IKM) is proposed to suppress the model mismatches and external disturbances of the 3-PRR CPM, where the unmodeled factors of the system are approximated through an online learning radical basis function neural network (RBFNN) and the external disturbances of the CPM is observed and compensated by a disturbance observer (DOB). Finally, a prototype of the 3-PRR CPM is manufactured, and experimental tests show the effectiveness of the proposed control scheme and the superiority of the 3-PRR CPM.

Influence of joints flexibility on overall stiffness of a 3-PRUP compliant parallel manipulator

The main aim of this paper is the influence assessment of the one of the most significant non-geometrical errors called compliance errors on accuracy of a 3-PRUP parallel robot. In this paper, a detailed and comprehensive study is performed to evaluate the effects of the joints and bodies flexibility on position accuracy of the robot. For this purpose, a continuous modeling method based on the energy method and Castigliano's 2nd theorem is presented. The capabilities of this method significantly reduce the limiting assumptions to obtain a highly accurate analytical stiffness model. Using the capabilities of this method, the compliance errors modeling of flexible bodies with complex geometry will be possible as well as the bending and torsional moments, shear forces and weight of all robot compliant modules can be considered as continuous loads. Using the concept of Wrench Compliant Module Jacobian, WCMJ, matrix and physical/structural properties matrix of each module, the compliance matrix of each flexible module is independently obtained. Using a computational algorithm, a comprehensive study is performed to obtain the contribution of the joints flexibility on the robot accuracy throughout the workspace. Finally, to evaluate the compliance errors boundaries throughout the workspace, the concept of General Compliance Error Range, GCER, is presented. The GCER represents a range between maximum and minimum accuracy of robot relative to the compliance errors.

A General Approach to the Large Deflection Problems of Spatial Flexible Rods Using Principal Axes Decomposition of Compliance Matrices

This paper presents a general discretization-based approach to the large deflection problems of spatial flexible links in compliant mechanisms. Based on the principal axes decomposition of structural compliance matrices, a particular type of elements, which relate to spatial six degrees-of-freedom (DOF) serial mechanisms with passive elastic joints, is developed to characterize the force-deflection behavior of the discretized small segments. Hence, the large deflection problems of spatial flexible rods can be transformed to the determination of static equilibrium configurations of their equivalent hyper-redundant mechanisms. The main advantage of the proposed method comes from the use of robot kinematics/statics, rather than structural mechanics. Thus, a closed-form solution to the system overall stiffness can be derived straightforwardly for efficient gradient-based searching algorithms. Two kinds of typical equilibrium problems are intensively discussed and the correctness has been verified by means of physical experiments. In addition, a 2DOF planar compliant parallel manipulator is provided as a case study to demonstrate the potential applications.

Synthesis and Evaluation of A High Precision 3D-Printed Ti6Al4V Compliant Parallel Manipulator

A novel 3D printed compliant parallel manipulator (CPM) with θX − θX − Z motions is presented in this paper. This CPM is synthesized using the beam-based method, a new structural optimization approach, to achieve optimized stiffness properties with targeted dynamic behavior. The CPM performs high non-actuating stiffness based on the predicted stiffness ratios of about 3600 for translations and 570 for rotations, while the dynamic response is fast with the targeted first resonant mode of 100Hz. A prototype of the synthesized CPM is fabricated using the electron beam melting (EBM) technology with Ti6Al4V material. Driven by three voice-coil (VC) motors, the CPM demonstrated a positioning resolution of 50nm along the Z axis and an angular resolution of ~0.3 “about the X and Y axes, the positioning accuracy is also good with the measured values of ±25.2nm and ±0.17” for the translation and rotations respectively. Experimental investigation also shows that this large workspace CPM has a first resonant mode of 98Hz and the stiffness behavior matches the prediction with the highest deviation of 11.2%. Most importantly, the full workspace of 10° × 10° × 7mm of the proposed CPM can be achieved, that demonstrates 3D printed compliant mechanisms can perform large elastic deformation. The obtained results show that CPMs printed by EBM technology have predictable mechanical characteristics and are applicable in precise positioning systems.

Kinetostatic modelling of a 3-PRR planar compliant parallel manipulator with flexure pivots

Flexure pivots are widely used in long stroke complaint parallel manipulators (CPMs) for their large deformation capacity. However the parasitic motion caused by the flexure pivots cannot be neglected as the rotation angle gets large, which will finally affect the absolute positioning accuracy of the manipulators. In this paper, a modelling approach to calculate the nonlinear kinetostatics of a long stroke planar CPM with flexure pivots is proposed, in which the parasitic motion of the flexure joints are taken into account. The displacement constraint equations and the static equilibrium equations of the CPM are established under the deformed configuration. Finite element analysis (FEA) shows the accuracy and efficiency of the nonlinear kinematic model (NLKM) for both the inverse and forward kinematic analysis. Moreover, the prediction accuracy of the CPM’s kinematic behavior has been improved more than 20 times by employing the NLKM compared with the conventional kinematic model (CKM).

A structure design method for compliant parallel manipulators with actuation isolation

Abstract. Since precise linear actuators of a compliant parallel manipulator suffer from their inability to tolerate the transverse motion/load in the multi-axis motion, actuation isolation should be considered in the compliant manipulator design to eliminate the transverse motion at the point of actuation. This paper presents an effective design method for constructing compliant parallel manipulators with actuation isolation, by adding the same number of actuation legs as the number of the DOF (degree of freedom) of the original mechanism. The method is demonstrated by two design case studies, one of which is quantitatively studied by analytical modelling. The modelling results confirm possible inherent issues of the proposed structure design method such as increased primary stiffness, introduced extra parasitic motions and cross-axis coupling motions.

Experimental validation of the kinematic design of 3-PRS compliant parallel mechanisms

In this paper, a procedure for the kinematic design of a 3-PRS compliant parallel manipulator of 3 degree of freedom is proposed. First, under the assumption of small displacements, the solid body kinematics of the 3-PRS has been studied, performing a comprehensive analysis of the inverse and forward kinematic problem, and calculating the rotations that the revolute and spherical flexure joints must perform. Then, after defining some design requirements and therefore the necessary displacements to fulfill, a design process based on the finite element calculations has been stablished, giving the necessary guidelines to reach the optimal solution on a 3-PRS compliant mechanism. Also, a prototype has been tested, using a coordinate measuring machine to verify its dimensions and the resulting displacements in the end effector and the actuated joints. Finally, those measurements have been compared with the FEM and the rigid body kinematics predictions, contrasting the validity of those two modelling approaches for the kinematic design of compliant mechanisms.

Design, modelling and analysis of a completely-decoupled XY compliant parallel manipulator

This paper deals with the design, modelling and analysis of a new XY compliant parallel manipulator (CPM). This XY CPM is composed of identical compound basic parallelogram modules as non-under-constrained compositional compliant modules, which is completely-kinetostatically decoupled. Based on an improved topology structure, a new XY CPM is proposed, which arranges each pair of prismatic joints on the base in two non-adjacent legs to be rigidly connected. Two analytical models, static approximate and static accurate, are derived for all translational displacements of rigid stages, and compared to a finite element analysis (FEA) model and experimental results for a case study. Detailed performance characteristics of the new XY CPM are analysed and compared with those of a corresponding XY CPM without the connection bars. The proposed new XY CPM has no under-constrained motion mass with a compact configuration. It is able to effectively constrain the parasitic rotation of all rigid stages and the cross-axis coupling of the motion stage. The cross-axis coupling motion of the actuation stage can be completely eliminated, and the lost motion between the actuation stage and the motion stage can be significantly reduced.

Design and nonlinear analysis of a 6-DOF compliant parallel manipulator with spatial beam flexure hinges

In this paper, a compliant parallel manipulator with six compliant limbs is proposed for micro positioning applications. The load–displacement model of a single compliant limb is established using a nonlinear closed-form spatial beam model. The inverse solution to the compliant parallel manipulator is then implicitly derived by applying load equilibrium to the moving platform. Finally, the compliant model of the limb and the implicit inverse kinematic solution of the manipulator are fully tested by FEA. Discrepancies between results of the presented models and the FEA are analyzed within planned workspaces. The validations demonstrate that accuracies of the proposed models are acceptable and can be improved by shrinking the planned workspace.

Design and Experimental Testing of an Improved Large-Range Decoupled XY Compliant Parallel Micromanipulator1

There is an increasing need for XY compliant parallel micromanipulators (CPMs) providing good performance characteristics such as large motion range, well-constrained cross-axis coupling, and parasitic rotation. Decoupled topology design of the CPMs can easily realize these merits without increasing the difficulty of controlling. This paper proposes an improved 4-PP model on the basis of a classical 4-PP model and both of them are selected for manufacturing and testing to verify the effectiveness of the improvement. It has shown from experimental results that there is a large improvement on the performances of improved 4-PP compliant parallel manipulator (CPM): large range of motion up to 5 mm × 5 mm in the unidirection in the dimension of 311 mm × 311 mm × 24 mm, smaller compliance fluctuation (only 36.63% of that of the initial 4-PP model), smaller cross-axis coupling (only 28.10% of that of the initial 4-PP model generated by a single-axis 5 mm actuation), smaller in-plane parasitic yaw (only 57.14% of that of the initial 4-PP model generated by double-axis 5 mm actuation).

Design of decoupled, compact, and monolithic spatial translational compliant parallel manipulators based on the position space

The initial version of this paper was presented at 2014 Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Tianjin, China. The conceptual design of three types of decoupled, compact, and monolithic XYZ compliant parallel manipulators (CPMs) is taken into account in this paper using the position space concept. The CPMs are termed CUBEs due to the shape of their compact structures. The position space of a compliant module is the combination of all permitted positions in an XYZ CPM system where the constraint of this compliant module in the XYZ CPM system remains unchanged when the position of the compliant module changes within the position space. The position of each compliant P joint is considered relative to its adjacent compliant joint/module rather than being considered in insolation. A design method for obtaining monolithic XYZ CPMs is proposed in terms of both the kinematic substitution and position spaces. Three types of monolithic XYZ CPMs are then demonstrated using the proposed method, with the help of three classes of kinematically decoupled three degrees of freedom translational parallel mechanisms. These monolithic XYZ CPMs include a 3- PPP (P: prismatic) XYZ CPM, a 3- PPPR (R: revolute) XYZ CPM, and a 3- PPPRR XYZ CPM. The position space concept can enable the system configurations to be designed based on the chosen optimization requirements, which include considerations such as compactness or minimization of parasitic rotations. This provides an efficient and systematic method to arrange the relative position between any two compliant joints/modules so that one can easily generate practical and useful configurations for XYZ CPMs. The resulting XYZ CPM is the most compact one when the fixed ends of the three actuated compliant P joints thereof overlap or are as close to each other as possible.

A constraint and position identification (CPI) approach for the synthesis of decoupled spatial translational compliant parallel manipulators

This paper introduces a screw theory based method termed constraint and position identification (CPI) approach to synthesize decoupled spatial translational compliant parallel manipulators (XYZ CPMs) with consideration of actuation isolation. The proposed approach is based on a systematic arrangement of rigid stages and compliant modules in a three-legged XYZ CPM system using the constraint spaces and the position spaces of the compliant modules. The constraint spaces and the position spaces are firstly derived based on the screw theory instead of using the rigid-body mechanism design experience. Additionally, the constraint spaces are classified into different constraint combinations, with typical position spaces depicted via geometric entities. Furthermore, the systematic synthesis process based on the constraint combinations and the geometric entities is demonstrated via several examples. Finally, several novel decoupled XYZ CPMs with monolithic configurations are created and verified by finite elements analysis. The present CPI approach enables experts and beginners to synthesize a variety of decoupled XYZ CPMs with consideration of actuation isolation by selecting an appropriate constraint and an optimal position for each of the compliant modules according to a specific application.

Stiffness characterization of a 3-PPR planar parallel manipulator with actuation compliance

This paper investigates the stiffness of a compliant planar parallel manipulator. Instead of establishing stiffness matrix directly for planar mechanisms, we adopt the modeling approach for spatial mechanisms, which allows us to derive two decoupled homogeneous matrices, corresponding to the translational and rotational stiffness. This is achieved by resorting to the generalized eigenvalue problem, through which the eigenscrew decomposition is implemented to yield six screw springs. The principal stiffnesses and their directions are then identified from the eigenvalue problem of the two separated submatrices. In addition, the influence of the nonlinear actuation compliance to the manipulator stiffness is investigated, and the established stiffness model is experimentally verified.

Conceptual designs of multi-degree of freedom compliant parallel manipulators composed of wire-beam based compliant mechanisms

This paper proposes conceptual designs of multi-degree(s) of freedom (DOF) compliant parallel manipulators (CPMs) including 3-DOF translational CPMs and 6-DOF CPMs using a building block based pseudo-rigid-body-model (PRBM) approach. The proposed multi-DOF CPMs are composed of wire-beam based compliant mechanisms (WBBCMs) as distributed-compliance compliant building blocks (CBBs). Firstly, a comprehensive literature review for the design approaches of compliant mechanisms is conducted, and a building block based PRBM is then presented, which replaces the traditional kinematic sub-chain with an appropriate multi-DOF CBB. In order to obtain the decoupled 3-DOF translational CPMs (XYZ CPMs), two classes of kinematically decoupled 3-PPPR (P: prismatic joint, R: revolute joint) translational parallel mechanisms (TPMs) and 3-PPPRR TPMs are identified based on the type synthesis of rigid-body parallel mechanisms, and WBBCMs as the associated CBBs are further designed. Via replacing the traditional actuated P joint and the traditional passive PPR/PPRR sub-chain in each leg of the 3-DOF TPM with the counterpart CBBs (i.e. WBBCMs), a number of decoupled XYZ CPMs are obtained by appropriate arrangements. In order to obtain the decoupled 6-DOF CPMs, an orthogonally-arranged decoupled 6-PSS (S: spherical joint) parallel mechanism is first identified, and then two example 6-DOF CPMs are proposed by the building block based PRBM method. It is shown that, among these designs, two types of monolithic XYZ CPM designs with extended life have been presented.