Pseudo-rigid-body Model Research Articles

The Interior Contact-Aided Rolling Element (I-CORE)

Abstract This work introduces an interior contact-aided rolling element (I-CORE) compliant mechanism that draws upon the concepts used for the contact-aided rolling element, cross axis flexural pivot, and pre-curved flexible beam. The I-CORE incorporates a bilinear compressive stiffness response with an initial tailorable stiffness governed by the flexural geometry, followed by a stiffness curve governed by the material stiffness at the contact point. The I-CORE mechanism can achieve 1 or 2 degrees of rotational freedom as well as 1 degree of translational freedom. The purpose of the present work was to introduce the I-CORE mechanism, as well as a pseudo-rigid-body replacement model (PRBM) of the I-CORE mechanism which was subsequently validated using both finite element analysis and bench top mechanical testing. A pseudo rigid body model was created for the I-CORE to simplify rapid adaptation of this mechanism to different design applications. This model was validated using both finite element analysis and benchtop mechanical testing under both compression and rotation loading conditions. Additionally, multiple configurations of the device were created and evaluated in order to test its sensitivity to certain design features including the flexure width, flexure thickness, and the radius of the rounded contact surfaces. It was found that the model is sensitive to the thickness of the flexures and that despite some limitations, the pseudo rigid body model is sufficiently accurate for initial design work. Some possible applications of the mechanism are proposed.

Design of FDM-printable tendon-driven continuum robots using a serial S-shaped backbone structure

Tendon-driven continuum robots (TDCR) are widely used in various engineering disciplines due to their exceptional flexibility and dexterity. However, their complex structure often leads to significant manufacturing costs and lengthy prototyping cycles. To cope with this problem, we propose a fused-deposition-modeling-printable (FDM-printable) TDCR structure design using a serial S-shaped backbone, which enables planar bending motion with minimized plastic deformation. A kinematic model for the proposed TDCR structure based on the pseudo-rigid-body model (PRBM) approach is developed. Experimental results have revealed that the proposed kinematic model can effectively predict the bending motion under certain tendon forces. In addition, analyses of mechanical hysteresis and factors influencing bending stiffness are conducted. Finally, A three-finger gripper is fabricated to demonstrate a possible application of the proposed TDCR structure.

Development of Modified Rhombus Amplifier of Positioning Stage Compliant Mechanisms

Abstract This paper presents a modified design of rhombus-type amplifier used in positioning stage compliant mechanism. The new rhombus design is introduced to enhance the performance of rhombus and the results are promising. The new approach of rhombus amplifier is based on mixing the concept of rhombus type with the idea of hinges (reduced section portions) existed in bridge type. The performance of the proposed design is investigated by the pseudo rigid body model and validated by the finite element analysis. The new rhombus design improves displacement amplification ratio by (22.3%) with slightly change in master mode natural frequency (1.3%). The pseudo rigid body model for new rhombus design is consistent with the ANSYS results, and hence can be used in modelling and analysis of the new rhombus design.

Design and performance evaluation of a novel parallel kinematic micromanipulator

The targeted placement of selected carbon nanotubes is associated with low throughput rates. Hence, this paper presents a novel stage design for the fully automatic assembly of carbon nanotube field-effect transistors (CNTFETs) via mechanical dry transfer. It constitutes the core module of an assembly machine for the precise deployment of nanotubes onto a prefabricated wafer. The mechanical dry transfer approach allows high level of carbon nanotube selectivity with the aim of enabling a high-volume fabrication of ultra-clean devices. The previous production rate of such ultra-clean devices is less than one per hour, which offers a high potential for improvement. The stage consists of a parallel kinematic mechanism (PKM) with 3 degrees of freedom (X2Y2C1) carrying two additional stacked axes (Z1C2). The PKM is suspended on a vacuum preloaded aerostatic bearing, voice coil motors (VCM) and flexure hinges. Together with a combination of high precision and resolution touch probe measurement systems close to its tool center point (TCP), high accuracy and low settling time can be achieved. Based on realistic manufacturing tolerances, a sensitivity analysis of the mechanism’s pseudo rigid body model (PRBM) suggests a theoretical closed-loop error below 0.15μm over the entire XY workspace of 20mm×2.8mm. Measurements of positioning motions show that the settling time can be decreased by the compensation of friction resistances inside the internal bearings of the VCMs.

A Rigid-Flexible Coupling Dynamic Model for Robotic Manta with Flexible Pectoral Fins

The manta ray, exemplifying an agile swimming mode identified as the median and paired fin (MPF) mode, inspired the development of underwater robots. Robotic manta typically comprises a central rigid body and flexible pectoral fins. Flexible fins provide excellent maneuverability. However, due to the complexity of material mechanics and hydrodynamics, its dynamics are rarely studied, which is crucial for the advanced control of robotic manta (such as trajectory tracking, obstacle avoidance, etc.). In this paper, we develop a multibody dynamic model for our novel manta robot by introducing a pseudo-rigid body (PRB) model to consider passive deformation in the spanwise direction of the pectoral fins while avoiding intricate modeling. In addressing the rigid-flexible coupling dynamics between flexible fins and the actuation mechanism, we employ a sequential coupling technique commonly used in fluid-structure interaction (FSI) problems. Numerical examples are provided to validate the MPF mode and demonstrate the effectiveness of the dynamic model. We show that our model performs well in the rigid-flexible coupling analysis of the manta robot. In addition to the straight-swimming scenario, we elucidate the viability of tailoring turning gaits through systematic variations in input parameters. Moreover, compared with finite element and CFD methods, the PRB method has high computational efficiency in rigid-flexible coupling problems. Its potential for real-time computation opens up possibilities for future model-based control.

A creative wide-frequency and large-amplitude vibration isolator design method based on magnetic negative stiffness and displacement amplification mechanism

Negative stiffness vibration isolation devices have attracted significant attention and research interest for their ability to enhance the low-frequency vibration isolation performance of passive vibration isolators. However, their narrow linear negative stiffness region has hindered their broader application. To address this limitation, a novel design method for vibration isolators is proposed, integrating the permanent magnetic negative stiffness device (NSD) with the displacement amplification mechanism (DAM). The paper begins by presenting a specific vibration isolator design method and analyzing the kinematic relationship. Next, a single-degree-of-freedom (SDOF) nonlinear equivalent model of the vibration isolator is established using the pseudo-rigid-body model (PRBM) approach. Numerical simulations are then conducted to obtain the transmissibility of the isolator. Furthermore, two isolators—one with the displacement amplification effect and another without—are manufactured, and experimental investigations are carried out. The experimental results confirm that the incorporation of the displacement amplification mechanism effectively expands the linear negative stiffness region of the negative stiffness magnet pair, enabling wide-frequency and large-amplitude vibration isolation. Notably, these experimental findings align well with the simulation results. In summary, this paper proposes a creative design method for wide frequency and large amplitude vibration isolators, and proves it through theoretical analysis and experimental validation.

Towards Robust and Effective Passive Compliance Design of End-Effectors for Robotic Train Fluid Servicing

Without mechanical compliance robots rely on controlled environments and precision equipment to avoid clashes and large contact forces when interacting with an external workpiece, e.g., a peg-in-hole (PiH) task. In such cases, passive compliance devices are used to reduce the insertion force (and in turn the robot payload) while guiding corrective motions. Previous studies in this field are limited to small misalignments and basic PiH geometries inapplicable to prevalent robotic and autonomous systems (RASs). In addition to these issues, our work argues that there is a lack of a unified approach to the development of passive compliance systems. To this end, we propose a higher-level design approach using robust engineering design (RED) methods. In a case study, we demonstrated this general approach with a Taguchi design framework, developing a remote centre compliant (RCC) end-effector for robotic train fluid servicing. For this specific problem, a pseudo-rigid-body model (PRBM) is suggested in order to save enormous computation time in design, modelling, and optimisation. Our results show that the compliant end-effector is capable of significantly reducing the insertion force for large misalignments up to 15 mm and 6 degrees.

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.

Design of a Compliant Vertical Micropositioning Stage Based on Lamina Emergent Mechanisms

This article presents the design and development of a novel compliant vertical micropositioning stage (VMS) with a millimeter stroke and compact profile. The concept design of the stage is inspired by lamina emergent mechanisms. The overall mechanism is fabricated using a plate-like material, which can realize an out-of-plane motion vertical to the mechanism plane. The proposed architectural design guarantees the advantages of the center platform in terms of millimeter stroke, high compactness, and low displacement coupling ratio. Analytical models of the stiffness and resonant frequency of the stage are established by resorting to pseudorigid-body model and Lagrangian equations. They are validated by conducting simulation study with finite element analysis. For performance testing, a prototype stage driven by a voice coil motor is fabricated. Experimental results indicate that the stage (size: 100 mm × 100 mm × 5 mm) delivers a working stroke of 5.58 mm, displacement coupling ratio of 0.31%, and resolution of 0.71 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> m. It exhibits more geometric advantages over existing piezoelectric-actuated vertical stages. In addition, it has superior vertical compactness, linearity, and simplicity compared to a planar mechanism with vertical input and vertical output. The concept design proposed in this article provides a new solution for the design and development of compliant VMS.

Development of a 3-DOF flexible micro-motion platform based on a symmetrical two-stage magnifying mechanism

A symmetrical 3-PRR flexible motion platform kinematic dynamic model is built. The natural frequency analysis of the platform is studied in this paper. Firstly, a flexible lever displacement two-stage amplification mechanism is proposed. A pseudo-rigid body model is established. A kinematics model is established by using the vector closed-loop method. The relationship between the input and output of the flexible lever displacement two-stage amplification mechanism is obtained. Secondly, based on the two-stage amplification mechanism of flexible lever displacement, a 3-PRR flexible motion platform is designed. The Jacobian matrix of input displacement of piezoelectric ceramics and output displacement of flexible motion platform is obtained by using the coordinate transformation method and vector equation closed-loop method. Considering the elastic potential energy of the flexible hinge and biaxial hinge, the Lagrange equation is established, and the expression of the natural frequency of the 3-PRR flexible motion platform is obtained. Finally, the correctness of the theoretical dynamic model is verified by theoretical calculation and finite element software simulation.

End Motion Path Analysis of Notched Variable Cross-section Flexible Manipulator Based on 3R Pseudo-Rigid Body Model

A wire-driven planar curved notched flexible manipulator with variable cross-section is designed. Based on the Bernoulli Euler beam equation, the end track and rotation angle of the notched flexible manipulator with variable cross-section after applying force is deduced theoretically. In addition, the 3R pseudo rigid body model is used to calculate the end track and rotation angle, and ABAQUS finite element analysis software is used for motion simulation. Then, MATLAB is used for post-processing to obtain the end track of the flexible manipulator. By comparing the track of the 3R pseudo rigid body model with the ABAQUS simulation track, the relative error is obtained. The conclusion proves that the 3R pseudo rigid body model can effectively simulate the end track of a notched flexible manipulator with variable cross-section after bending, which expands a new method for solving the complex end motion problem of a flexible manipulator with variable cross-section, and also broadens the applicable structure of the 3R pseudo rigid body model.

Additive Manufactured Piezoelectric-Driven Miniature Gripper

In several cases, it is desirable to have prototypes of low-cost fabrication and adequate performance. In academic laboratories and industries, miniature and microgrippers can be very useful for observations and the analysis of small objects. Piezoelectrically actuated microgrippers, commonly fabricated with aluminum, and with micrometer stroke or displacement, have been considered as Microelectromechanical Systems (MEMS). Recently, additive manufacture using several polymers has also been used for the fabrication of miniature grippers. This work focuses on the design of a piezoelectric-driven miniature gripper, additive manufactured with polylactic acid (PLA), which was modeled using a pseudo rigid body model (PRBM). It was also numerically and experimentally characterized with an acceptable level of approximation. The piezoelectric stack is composed of widely available buzzers. The aperture between the jaws allows it to hold objects with diameters lower than 500 μm, and weights lower than 1.4 g, such as the strands of some plants, salt grains, metal wires, etc. The novelty of this work is given by the miniature gripper’s simple design, as well as the low-cost of the materials and the fabrication process used. In addition, the initial aperture of the jaws can be adjusted, by adhering the metal tips in the required position.

Design, Modeling, and Evaluation of a Hybrid Driven Knee-Ankle Orthosis With Shape Memory Alloy Actuators

Abstract This paper demonstrates the mechanical design, analysis, and evaluation of a knee-ankle orthosis (KAO) for lower limb rehabilitation, including shape memory alloy (SMA) actuators and DC brushless motor actuators. First, the mechanical structure of the KAO is detailed, including the actuator system, transmission mechanism, monitoring device, and processing method of SMA actuators. Next, the dynamic model of SMA spring actuators in the phase transition process is established based on the thermal constitutive model of SMA. In addition, the dynamic output response of the knee joint under three working states is analyzed, and the rotation angle of SMA soft actuator during bending is described by pseudo rigid body model. Finally, the output of SMA actuator is preliminarily evaluated through experiments. The experimental results show that the maximum displacement of SMA spring actuator is 54.36 mm, the maximum restoring force during phase transformation is 4.14 N, and the maximum rotation angle of SMA soft actuator is 43.18 deg. The experimental results are consistent with the theoretical model.

Modelling of Tendon-Driven Continuum Robot Based on Constraint Analysis and Pseudo-Rigid Body Model

Quasi-static models of tendon-driven continuum robots (TDCR) require consideration of both the kinematic and static conditions simultaneously. While the Pseudo-Rigid Body (PRB-3R) model has been demonstrated to be efficient, existing works ignore the mechanical effect of the tendons such as elongation. In addition, the static equilibrium equations for the partially constrained tendons have been expressed in different forms within the literature. This leads to inconsistent simulation results which have not been validated by experimental data when external loads are applied. Furthermore, the inverse problem for solving the required inputs for a prescribed end effector pose has not been studied for the PRB-3R model. In this work, we introduce a new modelling approach based on constraint analysis (CA) of a multi-body system and Lagrange multipliers to systematically derive all the relevant governing equations required for a planar TDCR. This method can include tendon mechanics and efficiently solve for the direct and inverse kinetostatic models with either forces or displacements as the actuation inputs. We validate the proposed CA method using numerical simulation of a benchmark model and experimental data.

Zero-curvature deformation properties and 3R pseudo-rigid-body model of large-deflection Euler spiral beams

Euler spiral beams are curved in their undeflected state but can lay flat (zero-curvature deformed shape) when proper loads are applied at the end of the beams. This zero-curvature deformation property enables deployable compliant mechanisms to deform into compact configurations. However, the condition of zero-curvature deformation is still not clear and Euler spiral beams lack effective models for predicting deformation upon loading. To solve these problems, this paper first describes the shape of Euler spirals by using size-independent quantities. Based on Bernoulli–Euler beam theory, the large-deflection equations of Euler spiral beams are derived. The zero-curvature deformation properties of the Euler spiral beams with different boundary conditions are studied. The conditions for zero-curvature deformation are given. Meanwhile, a 3R pseudo-rigid-body model (PRBM) is proposed for Euler spiral beams. Analysis and design cases are used to verify the proposed model. The results will facilitate the use of the zero-curvature deformation property in developing compact deployable compliant mechanisms and the large-deflection analysis of Euler spiral beams.

Three-Dimensional Force Decouping-Sensing Soft Sensor with Topological Elastomer

Sensing the deformation of soft sensor elastomer can realize the flexible operation of soft robot and enhance the perception of human-computer interaction. The structural configuration of elastomer and its elastic deformation force transfer path are crucial for decoupling sensing and studying the sensing performance of three-dimensional force soft sensor. In this article, we present a theoretical method for soft sensor with three-dimensional force decoupling-sensing. First, the constraint types of parallel manipulator with three translational motion characteristics are analyzed and used to set the constraint conditions for topology optimization. In addition, the differential kinematic modeling method is adopted to establish the differential kinematic equation of the three translations parallel manipulator, which is used as a pseudo-rigid body model for sensor information perception. Second, combining the kinematic Jacobi matrix with solid isotropic material with penalization the (SIMP), the topological model is built for designing of sensor elastomer. We optimized the composition of the material and evaluate the model’s sensing capabilities. The results validate a elastomer of soft sensor for unity between structural stiffness and perceived sensitivity.

Design and Development of a New Piezoelectric-Actuated Biaxial Compliant Microgripper With Long Strokes

In this paper, a new piezoelectric-actuated biaxial compliant microgripper with long strokes is proposed for automatically gripping and rolling tiny rigid objects. In order to improve the working stroke and maintain a compact footprint, a counter-side distributed two-stage lever amplifier with parallelogram mechanism is introduced. Based on the pseudo-rigid body model, analytical models of the displacement amplification ratio, input stiffness, and natural frequency of the left- and right-sided gripper mechanism are established. Structural optimization and performance simulation of the proposed microgripper mechanism are carried out with finite element analysis simulation. A prototype microgripper has been fabricated for open-loop and closed-loop tests to verify its working capabilities. The clamping and rubbing experiments show that the designed microgripper can grasp and rub an optical fiber with the diameter of 200 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> for rolling over 45°. The developed microgripper has a promising application in precision micromanipulation fields such as optical fiber alignment. Note to Practitioners—This work is motivated by the requirement of designing a biaxial microgripper with both a large working stroke and compact structure for complex operation in optical fiber alignment. Through a series of open-loop and closed-loop experiments on the developed prototype, it is demonstrated that the proposed dual-axis microgripper has superior working performance. The clamping stroke and rubbing stroke of the gripper are <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$251.2~\mu \text{m}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$225.0~\mu \text{m}$ </tex-math></inline-formula> , respectively. The first two natural frequencies of the gripper are 350.63 Hz and 603.37 Hz, which correspond to the working modes of the right-side and left-side gripper mechanisms, respectively. The closed-loop experimental results show that the resolution of output displacement of the gripper is close to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.2~\mu \text{m}$ </tex-math></inline-formula> and the resolution of the clamping force is 3 mN. As compared with the reported microgrippers in previous work, the designed mechanism exhibits both a large working stroke and high resonant frequency for ensuring the reliability and rapidity of micromanipulation task.

A pseudo-rigid-body approach for determination of parasitic displacements of lumped compliant parallel-guiding mechanisms

Parasitic displacements of a guiding plate of a lumped compliant parallel-guiding mechanism are analyzed using the pseudo-rigid-body (PRB) approach. Small deformations of flexure hinges are assumed. Each flexure hinge of the compliant mechanism is modelled by the PRB model with 3-DOF (degrees of freedom). This model allows that axial deformation of the flexure hinge to be considered. The corresponding expressions in symbolic form for both translational and rotational parasitic displacements of the compliant mechanism are derived. The obtained expressions enable the analysis of the influence of various structural parameters as well as various types of flexure hinges on the parasitic displacements of the considered type of compliant parallel-guiding mechanisms.

On the numerical synthesis of a contact-aided branch hinge (CABH) with variable stiffness

Based on the mortise-and-tenon structure, this paper proposes a Contact-Aided Branch Hinge (CABH) with variable stiffness. The equivalent spring model (ESM) and the chain pseudo-rigid body model (CPRBM) of the hinge are derived respectively, and the applicability of these two models is analyzed by the finite element method and experimental tests. Therefore, a compliant-element-based dimensional synthesis (CEBDS) method is proposed. Taking the CABH as an example, three derivative mechanisms are synthesized to meet the complex working requirements. The finite element analysis (FEA) results are consistent with the kinematic requirements, which verifies the correctness and feasibility of the synthesis method. In addition, the prototypes are fabricated. The potential applications of the CABH in robotics and medical devices can be increased by combining multiple derivative mechanisms.

Design and Performance of a Spatial 6-RRRR Compliant Parallel Nanopositioning Stage

Piezoelectric actuators (PEAs) and compliant parallel mechanisms (CPMs) are advantageous for designing nanopositioning stages (NPSs) with multiple degrees of freedom (multi-DOFs). This paper proposes a new NPS that uses PEAs and CPMs with multiple spatial DOFs. First, the design of the mechanism is introduced. Six parallel kinematics revolute-revolute-revolute-revolute (RRRR) branched chains were used to create a 6-RRRR CPM for superior mechanical performance. Three in-plane and three out-of-plane chains were combined using a two-in-one structure to ensure fabrication feasibility. A two-in-one 6-RRRR CPM was employed to build the proposed NPS. Second, the mechanical performance was analyzed. High-efficiency finite-element modeling approaches were derived using the compliance-based matrix method (CMM) and a pseudo-rigid body model (PRBM). The model included both 6-RRRR CPM and NPS. The simulation results validated the static and dynamic performance, and the experimental results verified the kinematics. Based on the newly designed mechanism and verified mechanical performance, the proposed 6-RRRR NPS contributes to the development of spatial multi-DOF NPSs using PEAs and CPMs.