Parasitic Motion Research Articles

Parallel Continuum Delta: On the Performance Analysis of Flexible Quasi-Translational Robots

In the field of rigid parallel manipulators, the Delta parallel robot is one of the most popular choices in the industry due to its ability to adapt to a wide range of applications, particularly pick-and-place tasks. In this paper, the authors present novel designs of Delta-type continuum parallel manipulators with flexible bars, solving both their direct and inverse kinematics, as well as obtaining the associated workspace. The continuum parallel manipulators, unlike conventional robots, incorporate certain flexible elements, such as slender rods that make up the kinematic chains of the Delta manipulators proposed in this work. As a consequence of the flexibility of these rods, a purely translational movement will not be generated, since it is necessary to analyze the zones of the workspace where a parasitic motion related to the inclination of the moving platform compromises the task devised. In addition, an experimental prototype of the Keops-Delta continuum manipulator has been built, and several experimental tests have been carried out to validate the proposed theoretical model.

Kinematic calibration of a 5-DOF double-driven parallel mechanism with sub-closed loop on limbs

Abstract This paper proposes a kinematic calibration method of a novel 5-degree-of-freedom double-driven parallel mechanism with the sub-closed loop on limbs. At first, considering the introduction of a sub-closed loop significantly increased the complexity and difficulty of kinematic error modeling, an equivalent transformation method is proposed for the limb with a sub-closed loop. Then kinematic error model of the parallel mechanism is established based on the closed-loop vector method and parasitic motion analysis, which is verified by virtual prototype technology. Because the full kinematic error model is generally redundant, error parameter identifiability analysis is carried out by QR decomposition of the identification Jacobian matrix, and the redundant parameters are removed. Additionally, the Sequence Forward Floating Search algorithm is utilized to optimize measurement configurations to reduce the influence of measurement noise. Finally, with a laser tracker as the measuring device, numerical simulations and experiments are implemented to verify the proposed kinematic calibration method. The experiment results show that average position and orientation errors are reduced from 2.778 mm and 1.115° to 0.263 mm and 0.176°, respectively, within the prescribed workspace.

Kinematic analysis and experimental validation of a tripod parallel continuum manipulator

A tripod is a classical lower mobility parallel manipulator with three degrees of freedom. The usual geometrical constraints enforce a type of motion with two rotations on a horizontal plane and a translational motion in a vertical direction. These devices are applicable where a task requires a precise orientational motion. Nevertheless, some parasitic motions occur on the constrained degrees of freedom. Parallel Continuum Manipulators are devices where some elements have been replaced by slender rods whose deformation is source of mobility, allowing the elimination of some of the kinematic joints. This type of closed-loop compliant mechanisms have some features that can be useful in certain applications, e.g., the deformed state introduces a certain preload that avoids backslash, they have an inherent compliance that can be useful in some delicate tasks, and a certain control on wrench applied can be obtained. In this article, the comprehensive kinetostatic analysis of a tripod-type parallel continuum manipulator, 3 P ¯ FS , is explained and compared with that of its rigid counterpart 3 P ¯ RS . Workspace and singularity locus is obtained, positioning accuracy and wrench generation are evaluated experimentally.

Design, computational analysis and experimental study of a high amplification piezoelectric actuated microgripper

Increasing applications of compliant microgripper demands flexibility in working with a wide range of micro-objects which requires a large workspace, high precision motion, low parasitic motion, and satisfactory bandwidth control. To meet the requirement of pick and place manipulation tasks, a high amplification piezoelectric actuated microgripper is proposed and investigated in this paper. The high amplification of the microgripper is achieved using a compound amplifier. The compound amplifier is assisted to magnify the embedded piezoelectric actuator’s displacement. Two cascaded lever-type mechanisms are symmetrically connected with a bridge-type mechanism and form a three-stage amplification mechanism-based compound amplifier. Further, the four-bar parallelogram mechanisms are integrated with the third-stage displacement amplification mechanisms to linearize the output motion of the microgripper jaws. The characteristics of the microgripper were evaluated by computational analysis and validated using experimental investigations. Further, the design parameters are identified from the geometrical model of the individual displacement transmission mechanisms to perform a response surface optimization on the configured mechanism by the computational method. The design optimization of the microgripper resulted in a high displacement amplification ratio with a large workspace. The experimental investigations show that the designed microgripper is capable of achieving a high displacement amplification ratio of 34.5 and a total output displacement of 529.4 μm. Further, the characteristics of the microgripper such as motion resolution, and parasitic motion indicate that it will be able to perform high-precision micro-object grasping/releasing tasks.

Design and Analysis of a New Non-Parasitic Parallel Mechanism for 2T1R Motion

Abstract Two-limb parallel mechanisms (PMs) can have several advantages, such as avoidance of mechanical collision, fairly simple architecture, easy assembly, and large workspace. The paper presents a novel two-limb three-degree-of-freedom (DOF) PM with two translations and one-rotation (2T1R) that are actuated by two sliders. The manipulator runs with decoupled position and orientation with forward position solution without parasitic motions. The main topological characteristics of the PM such as position and orientation characteristic (POC), DOF, and coupling degree are analyzed. Second, using the kinematic modeling method based on topological characteristics, the unified symbolic forward and inverse position solutions of the two configurations of the PM (I and II) are derived together with the workspace, rotational capacity, and singularity. Finally, a comparative analysis regarding the kinematics of two configurations (I and II) is carried out to find an optimal design configuration I of a PM for 2T1R motion.

A novel stick-slip piezoelectric actuator is developed based on a parallelogram flexible drive mechanism to minimize shear force

A piezoelectric actuator utilizing a parallelogram flexible mechanism is proposed. This actuator harnesses the parasitic motion generated by a parallelogram driven by a piezoelectric element. It differs from existing actuation mechanisms based on a similar principle in that the piezoelectric stack is positioned on the outside of the parallelogram flexible mechanism, thus protecting it from shear forces. Finite element simulations and experiments confirm this. The analysis was conducted using theoretical analysis and finite element simulation. The optimal drive angle for the parallelogram flexible mechanism was determined through finite element simulation. Additionally, a prototype actuator and an experimental measurement system were developed to assess the operational performance of the proposed piezoelectric actuator. When the driving frequency is 475 Hz and the locking force is 5 N, the motion of the actuator achieves a maximum speed of 5.65 mm s−1 and a maximum horizontal load of up to 113 g; when the input frequency is 1 Hz and the input minimum starting voltage is 8 V, the minimum displacement resolution is 30 nm. A comparative analysis of experimental results, theoretical calculations, and finite element simulation results demonstrates the feasibility of the structural design.

Design and Analysis of a Symmetric Overconstrained Compliant Tilt/Tip Stage Based on a Hybrid Transmission Ratio Model

Abstract Developing a compliant mechanism that have potential in parasitism suppression and cross-axis decoupling is a major challenge to meet the requirement of spatial micro-/nano positioning. This work introduces a compliant tilt/tip stage design with a symmetric and overconstrained configuration that is equipped with four reverse bridge notch flexure amplifiers (RBNFAs) and five revolute notch flexure hinges as multiaxis decoupled structures. A hybrid transmission ratio model is developed to describe the mechanical behavior of this stage using elastic beam and pseudo-rigid-body theories. Finite element analysis (FEA) confirmed the analytical model results. A comprehensive study is performed based on FEA model to validate the influence of a particular configuration on parasitic motion and decoupling effect. A prototype stage is 3D printed and experimentally tested, which confirmed the predictions of the analytical hybrid model. In addition, further analysis was conducted to examine the static mechanical characteristics and parasitic behavior of the stage.

FlexDelta: A flexure-based fully decoupled parallel xyz positioning stage with a long stroke

Decoupled parallel xyz positioning stages with a large stroke are desired in high-speed and precise positioning applications. However, currently, such stages either have a short stroke or are unqualified in terms of the parasitic motion and coupling rate. In this paper, a novel flexure-based decoupled parallel xyz positioning stage (FlexDelta) is proposed and its conceptual design, modeling, and experimental investigation are conducted. First, the working principle of FlexDelta is introduced, followed by the design of its mechanism with flexure. Second, the stiffness model of flexure is established using matrix-based Castigliano’s second theorem, and the influence of its lateral stiffness on the stiffness model of FlexDelta is comprehensively investigated and then optimally designed. Finally, experiments were conducted using the fabricated prototype. The results reveal that the positioning stage features a centimeter stroke in three axes, with a coupling rate of less than 0.53% and parasitic motion of less than 1.72 mrad over the full range. Its natural frequencies are 20.8 Hz, 20.8 Hz, and 22.4 Hz for the x, y, and z axis, respectively. Multi-axis path-tracking tests were also performed, which validated its dynamic performance with a micrometer error.

A novel F-shaped linear guiding mechanism based compliant positioning stage with restricted parasitic motion

Positioning stages are significant in precision technologies under higher demands of operation and manufacturing. The performance of the selected guiding mechanisms is a decisive factor affecting the positioning accuracy of the stage. To restrict the parasitic motions along the degrees of constraint during motion transmission, a piezo-actuated compliant positioning stage with novel F-shaped linear guiding mechanisms (FLGM) is developed in this paper to enable a precision motion. With a simple and compact structure, a single FLGM can generate an approximate straight-line motion and enhance the transverse stiffness. Four FLGMs are arranged symmetrically in parallel to theoretically eliminate the parasitic motions, bringing in the merits of a low coupling rate and a wide bandwidth in the working direction. A stack piezoelectric actuator embedded in a forward bridge-type displacement amplifier is utilized to drive the guided stage. Analytical modeling of its kinematic, static, and dynamic characteristics is mathematically established by utilizing the pseudo-rigid-body method, compliance matrix method, and Lagrange’s equation. Finite element analysis is conducted and a fabricated prototype is experimentally tested, validating the veracity of the analytical models. Experimental results show that the parasitic motion is constrained by the guiding mechanism effectively with a coupling rate below 0.95%, and a high natural frequency of 856.9 Hz is achieved simultaneously.

ПРИДУШЕННЯ ПАРАЗИТНИХ КОМПОНЕНТІВ ДИНАМІКИ. МАГНІТОЛЕВІТУЮЧОГО ПОЇЗДА

Part of a maglev train’s partial motiones are parasitic - not useful, or malicious to the successful solution of it’s motor task. Under the conditions of unpredictable system’s disturbances, the requirement to preserve it’s purposefulness makes it rational the passive suppression of such motiones by the methods of the invariance theory. These methods are used to find conditions, the observance of which guarantees the independence of a suppressed coordinates from certain perturbations. Suppression can be done by achieving the invariance of coordinates that represent unwanted motiones with respect to ignored disturbances, as well as by isolating such coordinates from influences that destroy the desired invariance. Suppression of a parasitic motion components is carried out automatically. To do it, the structure and parameters of the system must be rationally selected, physically realizable and ensure its high dynamic qualities. It is convenient to choose the structure and parameters using the concepts of structural and parametric invariance. Their implementation is possible only if the two-channel principle is observed.

Development of a novel monolithic compliant Lorentz-force-driven XY nanopositioning system

A novel monolithic compliant Lorentz-force-driven XY nanopositioning system (MCLNS) is designed, analyzed, and experimentally assessed with the aim of high-resolution positioning across a large workspace. A double-symmetric Lorentz-force actuator (DSLA) with the benefits of zero friction, high thrust, and large stroke is proposed to generate the actuation force. Correspondingly, a monolithic four-prismatic parallel compliant mechanism (4P-PCM) is exploited to transmit the actuation motion to the central platform and minimize the parasitic motion. The unique integration of four DSLAs and one 4P-PCM make the proposed MCLNS possess compact structure and stable performance. The characterization of the MCLNS is formulated by a specially established analytical model and validated by finite-element analysis simulation and experimental tests. Experimental studies show that the workspace of the MCLNS prototype is large than 0.87 × 0.87 mm2 and the positioning resolution of the MCLNS prototype is better than 9 nm. By means of a nonlinear forward proportional integral derivative control strategy, the maximum contouring error of the MCLNS is maintained within 2.7% while tracking a 1257 μm s−1 circular trajectory.

Performance Analysis and Optimization Design of a Dual-Mode Reconfigurable Ankle Joint Parallel Rehabilitation Mechanism

In this study, a dual-mode reconfigurable parallel ankle joint rehabilitation mechanism is proposed to meet the needs of patients in different ankle rehabilitation stages. This mechanism can switch between the 1T2R (where R represents rotation and T represents translation) and 2T1R motion modes. The screw theory and the modified G-K formula were used to analyze and verify the degree of freedom of the mechanism. The non-parasitic motion characteristics were analyzed by examining the topological structure of the mechanism. An inverse kinematics model was established using the closed-loop vector method, and the mechanism’s singularity was analyzed based on the Jacobian matrix. The Jacobian matrix and the numerical method were used to compare and analyze the workspace index, the rotational dexterity index and the load capacity performance index before and after the introduction of branched chains with actuation redundancy. A particle swarm optimization algorithm was used to optimize the geometric dimensional parameters of the mechanism. The results show that the mechanism exhibits the characteristics of a parallel mechanism without parasitic motion in the two motion modes. Using branched chains with actuation redundancy can significantly improve the rotational dexterity and load capacity performance index, without affecting the workspace index. Compared to the original mechanism, the kinematic performance of the optimized mechanism is significantly improved. It is concluded that the proposed mechanism can meet the needs of ankle joint activity training in the 1T2R motion mode and the needs of ankle joint proprioception training in the 2T1R motion mode, which can better meet the needs of patients in different rehabilitation stages.

A piezo-actuated nanopositioning stage based on spatial parasitic motion principle

This paper presents the configuration, design, and characteristic analysis of a novel two-degree-of-freedom stick-slip nanopositioning stage based on the spatial parasitic motion principle. The planar-spatial evolution process is proposed to develop a compliant tripod mechanism by which the spatial parasitic motion can be generated for dual-axis actuation. An X-shaped hinge configuration is designed by freedom and constraint space analysis and introduced to the mechanism to improve the loading capacity of the nanopositioning stage. The actuation principle for driving along X- and Y-directions forward and backward is provided. The chain-based compliance matrix method is adopted for kinematic and static modeling of the compliant tripod mechanism. The dominant parameters are determined based on sensitivity analysis, and the performance of the stage is further studied by finite element method. The nanopositioning stage prototype is fabricated and assembled, and experimental investigations are conducted. The proposed nanopositioning stage has maximum velocities of 0.77 mm/s and 1.03 mm/s along X- and Y-direction, and a loading capacity of 5 kg can be achieved. Furthermore, the coarse-fine positioning experiments are carried out, and the positioning resolution is measured to be 18 nm.

Investigation of parasitic motion in 3-RRS parallel manipulator

This paper investigates the Parasitic Motion (PaMo) in a 3-Degree of Freedom (DoF) 3-RRS Spatial Parallel Manipulator (SPM) for its different link proportions. The position kinematic analysis of the manipulator is addressed initially. The PaMo is investigated and quantified as the percentage of Reachable Workspace (RW) for each link proportion at the tip of an End-Effector (EE), which is attached at the centroid of the moving platform. The PaMo which happens in the constrained DoF is an unwanted motion, and this entire process allows to know the total occupancy of PaMo in comparison to RW for each link proportion. The results are depicted with the aid of 3D plots of the reachable work-volume and the volume of PaMo for each link proportion. In order to evaluate the performance measure, a performance index called as the parasitic motion index ( PaMoI) is proposed. The PaMoI variation pattern is also discussed based on the obtained results. The conclusions are drawn at the end based on this investigation that also identify the provisions to minimize the PaMo.

Design and Analysis of Leaf Beam Single-Translation Constraint Compliant Modules and the Resulting Spherical Joints

Abstract A wire beam is a single-translation constraint along its axial direction. It offers many applications in compliant mechanisms, such as being a transmitting/decoupling element connected to a linear actuator and being a fundamental constitutive element to design complex compliant joints and mechanisms. It is desired to find an alternative leaf beam single-translation constraint to equal a wire beam in order to improve the manufacturability and robustness to external loading. In this paper, we propose and model a new single-translation constraint compliant module, I-shape leaf beam design, to compare with a corresponding L-shape leaf beam design reported in the literature. Two spherical (S) joints using three I-shape leaf beams and three L-shape leaf beams, respectively, are then analytically modeled and analyzed. Three key geometric parameters are adopted to thoroughly assess four performance indices of each S joint, including stiffness ratio, rotation radius error, coupling motion, and parasitic motion. It shows that the I-shape leaf beam–based S joint performance indices are generally 10 times better than those of the L-shape leaf beam–based S joint. For each S joint, the optimal parameters are found under the given conditions. Finally, experimental tests are carried out for a fabricated S joint prototype using the I-shape leaf beams, the results from which verify the accuracy of the proposed analytical model and the fabrication feasibility.

Design of a two-stage compliant asymmetric piezoelectrically actuated microgripper with parasitic motion compensation

The parasitic motion of microgripper jaws affect clamping accuracy. The parallelogram mechanism (PM) is often used as the last-stage of the multi-stage amplification mechanisms (AMs) because of its parallel clamping characteristics. However, the PM traditionally used in multi-stage AMs is asymmetrically stressed, which will affect its parallel clamping accuracy. Based on this, a PM with symmetrical forces is designed. The paper addresses the issue of parasitic motion in microgripper jaws and its impact on clamping accuracy, which is a relevant concern in micro-scale manipulation systems. The design of a symmetrically stressed PM for use in a two-stage asymmetric microgripper is a novel contribution to the field. Based on the minimum potential energy principle and the Castigliano's second theorem, the mechanical modeling of the mechanism's parasitic displacement, magnification and parallel clamping accuracy is carried out. Comparing the mechanical model, finite element analysis (FEA) and experimental measurement results, the values given by the three different methods are close. The method given in this paper to improve the clamping accuracy of the mechanism is also applicable to similar compliant mechanisms.

Six-Bar Linkages With Compliant Mechanisms for Programmable Mechanical Structures

Abstract Programmable mechanical structures are formed by autonomous and adaptive cells and can reproduce meshes known from the finite element method. Furthermore, they can change their structure not only through morphing, but also by self-reconfiguration of the cells. A crucial component of the cells, which can preserve the underlying geometry of a triangular mesh, are six-bar linkages. The main part of the present contribution concerns the six-bar linkages as a fully 3D-printable compliant mechanism where each revolute joint of the six-bar linkage is replaced with a notch flexure hinge with the circular contour. The utilization of notch flexure hinges presents two significant drawbacks. First, notch flexure hinges do not maintain the center of rotation. Second, although compliance is an inherent and desirable characteristic of flexural hinges, it gives rise to secondary or parasitic motion. The compliance subsequently lead to alterations in the underlying geometry of a triangular mesh. For self-reconfiguration of the cells, an efficient model is needed to predict the positioning errors. Therefore, the flexure hinge is represented by three distinct models, namely a finite element model, a beam model, and a simplified linearized model based on translational and rotational spring elements. These models are compared and evaluated in succession first to identify the parameters of the simplified model and later on, the simplified model is used to show the deviations of a medium-scaled programmable structure with respect to the idealized behavior. The current work brings us closer to both the development of programmable mechanical structures and the prediction of positioning errors during self-reconfiguration.

Motion Performance Study of 2UPR-1RPS/2R Hybrid Robot Based on Kinematics, Dynamics, and Stiffness Modeling

Abstract The unique structural characteristics of hybrid robots, such as few degrees-of-freedom (DOF) and redundant constraints, lead to a series of challenges in the establishment of theoretical models. However, these theoretical models are indispensable parts of motion control. Therefore, this paper focuses on establishing the kinematics, dynamics, and stiffness models for an Exechon-like hybrid robot, which are then used for error compensation and velocity planning to improve the robot’s motion performance. First, the kinematic model is derived through intermediate parameters and the kinematics equivalent chains. By analyzing the parasitic motion due to few DOF, the redundant equations in the model are eliminated to obtain the solution of inverse kinematics. Second, based on the beam element, the optimal equivalent configuration of the moving platform which connects the parallel part and serial part is determined, and then an entire equivalent structure of the robot is formed. It helps establish the stiffness model by using the matrix structure analysis method. Next, the dynamic model is established by combining the Newton–Euler method with co-deformation theory to solve the underdetermined dynamic equations caused by redundant constraints. Finally, the compensation method is designed based on the stiffness model and kinematic model to improve the end positioning accuracy of the robot; the velocity planning algorithm is designed based on the dynamic model and kinematic model to enhance the smoothness of the robot motion. The methods proposed in this paper are also of referential significance to other Exechon-like hybrid robots.

Design of a Novel Flexible Spherical Hinge and Its Application in Continuum Robot

Abstract Compliant mechanisms, which can be integrally machined and without assembly, are well suited as joints for continuum robots (CRs), but how to incorporate the advantages of the compliant mechanism into the arm design is a key issue in this work. In this paper, a novel type of flexible spherical-hinged (FSH) joint composed of tetrahedron elements with a fixed virtual remote center of motion (RCM) at the bottom is proposed, and then extended to the CR and end-effector. In the arm design, the error compensation principle is used to offset the parasitic motion of the CR under external load (pressure and torque) and improve the bending and torsional isotropy of the arm through different series combinations, and then the stiffness model of the FSH joint and the statics model of the CR are developed using the 3D chain pseudo-rigid body model (3D-CPRBM) and tested. The results show that the 3D-CPRBM can effectively predict the deformation of the FSH joint and the CR. Moreover, the maximum standard deviation of the bending angle of the FSH joint in each direction is only 0.26 deg, the repeatable positioning accuracy of the CR can reach 0.5 deg, and the end-effector has good gripping ability and self-adaptive capability.

Comparison of 3-[PP]S-Y Spatial Parallel Manipulators based on link tolerances and parasitic motions

This paper strives to compare between two 3-Degree of Freedom (DoF) Spatial Parallel Manipulators (SPMs) belonging to the 3-[PP]S-Y family viz., 3-RRS and 3-RPS. The planes of the manipulators belong to this family are arranged in a Y-pattern such that they intersect at 120° in a common line. Both of these manipulators have a common motion of 1T2R (T: Translation, R: Rotation) type, and their comparison in this work is carried out by considering the effects of link tolerance and parasitic motions, for the same task space and End-Effector (EE) positions. Their respective position kinematic analysis is discussed initially. The comparison is then carried out by considering two different cases based on the sizes of the moving platform relative to the base platform. The errors are quantified and all the comparative results are represented with the aid of plots. The conclusions are summarized at the end that allow to know the better of the two manipulators for the given effects.