Pseudo-rigid-body Models Research Articles

High-order kinematics of uniform flexures

In the last decades, many efforts have been made to analyze and model small and large deflections of flexures, considering complex load cases and different solution techniques. However, few investigations focused on the kinematic aspects related to the deflection analysis of the flexible elements, and limited the study to the second-order kinematics. In this paper, an analytical formulation based on the instantaneous geometric invariants is developed to give deep kinematic insight, up to the fourth order, into the motion generated by the deflections. The problem is addressed from a geometrical point of view, defining the fundamental geometric entities that characterize the motion, that are inflection circle, cubic of stationary curvature and its derivative, Ball’s point, and Burmester’s points. Thanks to these special points on the plane, straight and circular paths can be approximated to the third and to the fourth order, respectively. The proposed formulation defines the geometric characteristics of flexures with different curvatures. An application regarding the definition of pseudo-rigid body models is discussed. Finite element simulations are performed to validate the results.

Design and optimization of a decoupled serial constant force microgripper for force sensitive objects manipulation

AbstractTo address coupling motion issues and realize large constant force range of microgrippers, we present a serial two-degree-of-freedom compliant constant force microgripper (CCFMG) in this paper. To realize a large output displacement in a compact structure, Scott–Russell displacement amplification mechanisms, bridge-type displacement amplification mechanisms, and lever amplification mechanisms are combined to compensate stroke of piezoelectric actuators. In addition, constant force modules are utilized to achieve a constant force output. We investigated CCFMG’s performances by means of pseudo-rigid body models and finite element analysis. Simulation results show that the proposed CCFMG has a stroke of 781.34 ${\unicode[Times]{x03BC}}\mathrm{m}$ in the X-direction and a stroke of 258.05 ${\unicode[Times]{x03BC}}\mathrm{m}$ in the Y-direction, and the decoupling rates in two directions are 1.1% and 0.9%, respectively. The average output constant force of the clamp is 37.49 N. The amplification ratios of the bridge-type amplifier and the Scott–Russell amplifier are 7.02 and 3, respectively. Through finite element analysis-based optimization, the constant force stroke of CCFMG is increased from the initial 1.6 to 3 mm.

Pseudo-Rigid-Body Dynamic Models for Design of Compliant Members

Abstract Movement in compliant mechanisms is achieved, at least in part, via deformable flexible members, rather than using articulating joints. These flexible members are traditionally modeled using finite element analysis (FEA)-based models. In this article, an alternative strategy for modeling compliant cantilever beams is developed with the objectives of reducing computational expense and providing accuracy with respect to design optimization solutions. The method involves approximating the response of a flexible beam with an n-link/m-joint pseudo-rigid-body dynamic model (PRBDM). Traditionally, static pseudo-rigid-body models (PRBMs) have shown an approximation of compliant elements using two or three revolute joints (2R/3R-PRBM). In this study, a more general nR-PRBDM model is developed. The first n resonant frequencies of the PRBDM are matched to exact or FEA solutions to approximate the response of the compliant system and compared with existing PRBMs. PRBDMs can be used for co-design studies of flexible structural members and are capable of modeling large deflections of compliant elements. We demonstrate PRBDMs that show dynamically accurate response for a random geometry cantilever beam by matching the steady-state and frequency response, with dynamical response accuracies up to 10% using a 5R-PRBDM.

Introducing mass parameters to Pseudo–Rigid–Body models for precisely predicting dynamics of compliant mechanisms

In this paper, we introduce non-dimensional mass property parameters to the classical PRB model for accurately predicting the dynamics of the compliant mechanisms. Given extensive work on PRB models accurately modeling the statics and kinematics of compliant mechanisms, few works have investigated on their accuracy in predicting dynamics. Here we study the fundamental natural frequency of two typical mechanisms: fixed-root and pinned-root compliant parallel-guiding mechanisms (CPGM). First, analytical expressions of the natural frequencies of the continuum models are derived according to the vibration mechanics. Second, theoretical expressions of the natural frequencies of the parameterized PRB model are obtained from their dynamics equations. Third, the mass property parameters are optimized to minimize the error between the continuum models and the parameterized PRB model. We conclude that the optimized PRB models can well predict the dynamics especially the low input signal frequency or mass of beams are not negligible. These new PRB models can significantly improve computational efficiency in dynamics simulation of compliant mechanisms.

Enhanced control of a flexure-jointed micromanipulation system using a vision-based servoing approach

This paper describes a high-precision motion control implementation for a flexure-jointed micromanipulator. A desktop experimental motion platform has been created based on a 3RUU parallel kinematic mechanism, driven by rotary voice coil actuators. The three arms supporting the platform have rigid links with compact flexure joints as integrated parts and are made by single-process 3D printing. The mechanism overall size is approximately 250x250x100 mm. The workspace is relatively large for a flexure-jointed mechanism, being approximately 20x20x6 mm. A servo-control implementation based on pseudo-rigid-body models (PRBM) of kinematic behavior combined with nonlinear-PID control has been developed. This is shown to achieve fast response with good noise-rejection and platform stability. However, large errors in absolute positioning occur due to deficiencies in the PRBM kinematics, which cannot accurately capture flexure compliance behavior. To overcome this problem, visual servoing is employed, where a digital microscopy system is used to directly measure the platform position by image processing. By adopting nonlinear PID feedback of measured angles for the actuated joints as inner control loops, combined with auxiliary feedback of vision-based measurements, the absolute positioning error can be eliminated. With controller gain tuning, fast dynamic response and low residual vibration of the end platform can be achieved with absolute positioning accuracy within ±1 micron.

Speeding Up Topology Optimization of Compliant Mechanisms With a Pseudorigid-Body Model

This paper seeks to speed up the topology optimization using a pseudorigid-body (PRB) model, which allows the kinetostatic equations to be explicitly represented in the form of nonlinear algebraic equations. PRB models can not only accommodate large deformations but more importantly reduce the number of variables compared to beam theory or finite element methods. A symmetric 3R model is developed and used to represent the beams in a compliant mechanism. The design space is divided into rectangular segments, while kinematic and static equations are derived using kinematic loops. The use of the gradient and hessian of the system equations leads to a faster solution process. Integer variables are used for developing the adjacency matrix, which is optimized by a genetic algorithm. Dynamic penalty functions describe the general and case-specific constraints. The effectiveness of the approach is demonstrated with the examples of a displacement inverter and a crimping mechanism. The approach outlined here is also capable of estimating the stress in the mechanism which was validated by comparing against finite element analysis. Future implementations of this method will incorporate other pseudorigid-body models for various types of compliant elements and also try to develop multimaterial designs.

A general and efficient multiple segment method for kinetostatic analysis of planar compliant mechanisms

In the past decades, numerous mechanics models and mathematical formulations have been developed for kinetostatic analysis of compliant mechanisms. However, it is rather tedious and error-prone to derive analysis equations based on these models. In this work, we present a general kinetostatic analysis framework for planar compliant mechanisms in which 2D beams can be represented by multiple segments of three commonly used models: beam-constraint-model (BCM), linear Euler–Bernoulli beam and pseudo-rigid-body models (PRBM). The framework is developed such that any beam model with a closed-form energy equation can be integrated without the deep understanding of the proposed scheme. The static equilibrium equations are automatically derived based on kinematic vector loop and solved based on minimization of total potential energy. Since the PRBM only returns the tip deflection, we have developed a procedure for calculating strain energy, actual beam shape and bending stresses from the tip deflection. This framework has been implemented DAS2D, an open-source object oriented software.

A Practical Method for Determining the Pseudo-rigid-body Parameters of Spatial Compliant Mechanisms via CAE Tools

Compliant Mechanisms (CMs) are employed in several applications requiring high precision and reduced number of parts. For a given topology, CM analysis and synthesis may be developed resorting to the Pseudo-Rigid Body (PRB) approximation, where flexible members are modelled via a series of spring-loaded revolute joints, thus reducing computational costs during CM simulation. Owing to these considerations, this paper reports about a practical method to determine accurate PRB models of CMs comprising out-of-plane displacements and distributed compliance. The method leverages on the optimization capabilities of modern CAE tools, which provide built-in functions for modelling the motion of flexible members. After the validation of the method on an elementary case study, an industrial CM consisting of a crank mechanism connected to a fully-compliant four-bar linkage is considered. The resulting PRB model, which comprises four spherical joints with generalized springs mounted in parallel, shows performance comparable with the deformable system.

Parameter Optimization of Pseudo-Rigid-Body Models of MRI-Actuated Catheters.

Simulation and control of a system containing compliant mechanisms such as cardiac catheters often incur high computational costs. One way to reduce the costs is to approximate the mechanisms with Pseudo-Rigid-Body Models (PRBMs). A PRBM generally consists of rigid links connected by spring-loaded revolute joints. The lengths of the rigid links and the stiffnesses of the springs are usually chosen to minimize the tip deflection differences between the PRBM and the compliant mechanism. In most applications, only the relationship between end load and tip deflection is considered. This is obviously not applicable for MRI-actuated catheters which is actuated by the coils attached to the body. This paper generalizes PRBM parameter optimization to include loading and reference points along the body.

Extension Effects in Compliant Joints and Pseudo-Rigid-Body Models

Compliant members come in a variety of shapes and sizes. While thin beam flexures are commonly used in this field, they can be replaced by soft members with lower aspect ratio. This paper looks to study the behavior of such elements by analyzing them from the view of beam theory for 2D cases. A modified version of the Timoshenko beam theory is presented which incorporates extension and Poisson's effects. The utility and validity of the new approach are demonstrated by comparing against Euler–Bernoulli beam theory, Timoshenko beam theory, and finite-element analysis (FEA). The results from this are then used to study the performance of pseudo-rigid-body models (PRBMs) for the analysis of low aspect ratio soft compliant joints for 2D quasi-static applications. A parallel-guiding mechanism comprised of similar compliant elements is analyzed using the new results to validate the contribution of this work.

A Three-Spring Pseudorigid-Body Model for Soft Joints With Significant Elongation Effects

Compliant mechanisms achieve motion utilizing deformation of elastic members. However, analysis of compliant mechanisms for large deflections remains a significant challenge. In this paper, a three-spring revolute–prismatic–revolute (RPR) pseudorigid-body (PRB) model for short beams used in soft joints made of elastomer material is presented. These soft joints differ from flexure-based compliant joints in which they demonstrate significant axial elongation effects upon tip loadings. The traditional PRB models based on long thin Euler beams failed to capture this elongation effect. To overcome this difficulty, a model approximation based on the Timoshenko beam theory has been derived. These equations are utilized to calculate the tip deflection for a large range of loading conditions. An optimization process is then carried out to determine the optimal values of the parameters of the PRB model for a large range of tip loads. An example based on a robotic grasper finger is provided to demonstrate how the model can be used in analysis of such a system. This model will provide a simple approach for the analysis of compliant robotic mechanisms.

Pseudorigid-Body Models of Compliant DNA Origami Mechanisms

In this paper, we introduce a strategy for the design and computational analysis of compliant DNA origami mechanisms (CDOMs), which are compliant nanomechanisms fabricated via DNA origami self-assembly. The rigid, compliant, and flexible parts are constructed by bundles of many double-stranded DNA (dsDNA) helices, bundles of a few dsDNA helices or a single dsDNA helix, and single-stranded DNA (ssDNA) strands, respectively. Similar to its macroscopic counterparts, a CDOM generates its motion via deformation of at least one structural member. During the motion, strain energy is stored and released in the compliant components. Therefore, these CDOMs have the advantage of suppressing thermal fluctuations due to the internal mechanical energy barrier for motion. Here, we show that classic pseudorigid-body (PRB) models for compliant mechanism are successfully employed to the analysis of these DNA origami nanomechanisms and can serve to guide the design and analysis method. An example of compliant joint and a bistable four-bar CDOM fabricated with DNA origami are presented.

Adaptive pseudo-rigid-body model for generalized cross-spring pivots under combined loads

Generalized cross-spring pivots (CSPs) are widely used as revolute joints in precision machinery. However, pseudo-rigid-body (PRB) models cannot capture the parasitic motions of a generalized CSP exactly under combined loads; moreover, the characteristic parameters used in PRB methods must be recomputed using optimization techniques. In this study, we develop two simple and accurate PRB models for generalized CSPs. First, a PRB method for a beam is developed based on the beam constraint model and the instantaneous center model, where the beam is modeled as two rigid links joined at a pivot via a torsion spring. Subsequently, two PRB models of the generalized CSP, comprising a four-bar model for accuracy and a pin-joint model for stiffness, are constructed based on a kinematic analysis using the proposed PRB method. A deflection characteristic analysis is then conducted to determine the relationship between the proposed model and the existing models. Finally, the PRB models for the pivot under the action of combined loads are validated via finite element analysis. The error evaluation indicates that the proposed PRB models are more accurate than the results from existing methods. The PRB models proposed here can be used in parametric design of compliant mechanisms.

A parameter optimization framework for determining the pseudo-rigid-body model of cantilever-beams

This paper focuses on developing a framework for determining the optimal pseudo-rigid-body (PRB) model of 2D cantilever beams. PRB models are commonly used in design and analysis of compliant mechanisms since they significantly reduce the number of degrees of freedom compared with the finite element approach. Although a number of PRB models are available in literature, there is not a unified method to determine the most suitable pseudo-rigid-body model for a specific application. In this work, we first study a modified Timoshenko beam equation which accommodates shear forces and axial deformation. The numerical solution to the Timoshenko beam equation provides a baseline for comparing various models. A novel concept of “PRB matrix” is proposed for representing topologies of all PRB models in a uniform way. The optimal set of kinematic parameters (characteristic lengths and spring constants) of PRB models are determined by minimizing the error of tip deflection and comparing with the solution of the Timoshenko beam equation. To validate this formulation, we compare the results with existing PRB models and obtained equivalent if not a more accurate set of PRB parameters. At last, a case study of a compliant slider mechanism is provided to demonstrate the accuracy of two PRB models in this particular application.

Simplified PRBMs of spatial compliant multi-beam modules for planar motion

Abstract. PRBMs (pseudo-rigid-body models) have been becoming important engineering technologies/methods in the field of compliant mechanisms to simplify the design and analysis through the use of the knowledge body of rigid-body mechanisms coupling with springs. This article addresses the PRBMs of spatial multi-beam modules for planar motion, which are composed of three or more symmetrical wire/slender beams parallel to each other where the planar twisting DOF (degree of freedom) is assumed to be very small for specific applications/loading conditions. Simplified PRBMs are firstly proposed through replacing each beam in spatial multi-beam module with a rigid-body link plus two identical spherical joints at its two ends. The characteristics factor, bending stiffness and twisting stiffness for the spherical joint are determined. Load-displacement equations are then derived for a class of spatial multi-beam modules and general spatial multi-beam modules using the virtual work principle and kinematic relationships. Finally, nonlinear FEA (finite element analysis) is employed with comparisons with the PRBMs. The present PRBMs have shown the ability to predict the primary nonlinear constraint characteristics such as load-stiffening effect, cross-axis coupling in the two primary translational directions and buckling load.

Analysis of a Fixed-Guided Compliant Beam With an Inflection Point Using the Pseudo-Rigid-Body Model Concept

This paper provides an efficient method of analysis for a fixed-guided compliant beam with an inflection point, subjected to beam end load or displacement boundary conditions, or a combination thereof. To enable this, such a beam is modeled as a pair of well-established pseudo-rigid-body models (PRBMs) for fixed-free compliant beam segments. The analysis procedure relies on the properties of inflection in developing the necessary set of parametric, static equilibrium and compatibility equations for solution. The paper further discusses the multiplicity of possible solutions, including displacement configurations, for any two specified beam end displacement boundary conditions, depending on the locations and types of the effecting loads on the beam to meet these boundary conditions. A unique solution may exist when a third beam end displacement boundary condition is specified; however, this selection is not unconditional. A concept of characteristic deflection domain is proposed to assist with the selection of the third boundary condition to yield a realistic solution. The analysis method is also used to synthesize a simple, fully compliant mechanism utilizing the fixed-guided compliant segments.

A Numerical Method for Position Analysis of Compliant Mechanisms With More Degrees of Freedom Than Inputs

An underactuated or underconstrained compliant mechanism may have a determined equilibrium position because its energy storage elements cause a position of local minimum potential energy. The minimization of potential energy (MinPE) method is a numerical approach to finding the equilibrium position of compliant mechanisms with more degrees of freedom (DOF) than inputs. Given the pseudorigid-body model of a compliant mechanism, the MinPE method finds the equilibrium position by solving a constrained optimization problem: minimize the potential energy stored in the mechanism, subject to the mechanism’s vector loop equation(s) being equal to zero. The MinPE method agrees with the method of virtual work for position and force determination for underactuated 1-DOF and 2-DOF pseudorigid-body models. Experimental force-deflection data are presented for a fully compliant constant-force mechanism. Because the mechanism’s behavior is not adequately modeled using a 1-DOF pseudorigid-body model, a 13-DOF pseudorigid-body model is developed and solved using the MinPE method. The MinPE solution is shown to agree well with nonlinear finite element analysis and experimental force-displacement data.

An effective pseudo-rigid-body method for beam-based compliant mechanisms

Beam-based compliant mechanisms have been widely used in many precision applications due to their large-displacement characteristic. Meanwhile, a simple and accurate analytical tool is no doubt useful to assist design and optimize these mechanisms. The pseudo-rigid-body (PRB) model is such a good alternative. The PRB model in a traditional sense, however, is too approximate to calculate some precise characteristics such as center-shift of a flexural joint. In this paper, a new method is proposed to easily establish simple and accurate PRB models for a variety of beam-based compliant mechanisms. As is well known, for most of the beam-based compliant mechanisms, there is usually an instantaneous center of rotation (ICR) on each beam, and the beam can be considered as a fixed-guided beam with one end fixed while the other end moves along a certain curve that usually is an approximate arc around the ICR. In this case, the behavior of beams can be imitated by a rigid bar with two pin-joints. One of the pin-joints is directly added at the end of the rigid bar. The position of the other is determined by the characteristic radius factor, and a torsional spring is added at the position to represent the beam's stiffness. The modeling method is used to analyze two commonly used beam-based compliant mechanisms as examples, i.e. a parallel linear spring stage and a cross strip pivot. The performances of precision, stiffness and stroke for the two mechanisms are investigated, and the method is validated by comparing the results with other research work presented in the literatures. Both cases show the method is intuitive and accurate, and it will be useful in the early phases of the design for beam-based compliant mechanisms.

Modeling of Flexural Beams Subjected to Arbitrary End Loads

The analysis of compliant mechanisms is often complicated due to the geometric nonlinearities which become significant with large elastic deflections. Pseudo rigid body models (PRBM) may be used to accurately and efficiently model such large elastic deflections. Previously published models have only considered end forces with no end moment or end moment acting only in the same direction as the force. In this paper, we present a model for a cantilever beam with end moment acting in the opposite direction as the end force, which may or may not cause an inflection point. Two pivot points are used, thereby increasing the model’s accuracy when an inflection point exists. The Bernoulli-Euler beam equation is solved for large deflections with elliptic integrals, and the elliptic integral solutions are used to determine when an inflection point will exist. The beam tip deflections are then parameterized using a different parameterization from previous models, which renders the deflection paths easier to model with a single degree of freedom system. Optimization is used to find the pseudo rigid body model which best approximates the beam deflection and stiffness. This model, combined with those models developed for other loading conditions, may be used to efficiently analyze compliant mechansims subjected to any loading condition.