Cartesian Stiffness Matrix Research Articles

The decoupling of the Cartesian stiffness matrix in the design of microaccelerometers

The 6×6 Cartesian stiffness matrix obtained through finite element analysis for compliant mechanical structures may lead to spurious coupling that stems from discretization error. The coupling may lead, in turn, to inaccurate results of the translational and rotational displacement analysis of the structure, for which reason a reliable decoupling technique becomes essential. In this paper, the authors resort to a decoupling technique of the Cartesian stiffness matrix, reported elsewhere, which is applied to the stiffness matrix of a class of accelerometers. In doing this, the generalized eigenvalue problem is first recalled as a powerful tool that is pertinent to the design task at hand (Ding and Selig in Int. J. Mech. Sci. 46(5): 703–727, 2004). The decoupled submatrices are then investigated by means of eigenvalue analysis. As a consequence, the translational and rotational stiffness matrices can be analyzed independently. Meanwhile, the decoupled stiffness matrices reveal compliance along the sensitive axes and high off-axis stiffness, thereby satisfying the ultimate design objectives for microaccelerometers with isotropic, monolithic structure.

Cartesian Stiffness Matrix Mapping of a Translational Parallel Mechanism with Elastic Joints

This paper is devoted to calculating the Cartesian stiffness matrix of a translational parallel manipulator with elastic joints. The calculation takes into account the contribution of the Jacobian variation because of the change of manipulator configuration due to the elasticity and it covers the entire theoretical workspace of the manipulator. Three kineto-static adimensional indices are proposed to measure the response of the manipulator in terms of stiffness.

On the Lagrangian and Cartesian Stiffness Matrices of Parallel Mechanisms with Elastic Joints

In this paper an analytical formulation is proposed of the Lagrangian and Cartesian stiffness matrices for parallel mechanisms with rigid bodies and elastic joints. The formulation is general, as it is based on the development of the principle of virtual work and on the definition of the stiffness matrices. A relationship is also obtained between the Lagrangian and Cartesian stiffness matrices. Each contribution to the stiffness matrices is explicitly expressed in order to gain insight into their physical meaning and mathematical nature. In the paper, three numerical examples are presented. Indeed, the stiffness matrices are calculated for three planar mechanisms. The computation method exploits the multiple closed-chain architecture in parallel architectures. For this reason, it can be performed without encountering practical difficulties.

Workspace of parallel kinematics machines with minimum stiffness limits: Collinear stiffness value based approach

Stiffness features of the parallel kinematics machines (PKM) are considered. To overcome the difficulties caused by heterogeneous units of measurement of the Cartesian stiffness matrix entries, a new performance index – the collinear stiffness value (CSV) – is applied. The CSV gives natural stiffness evaluation in both regular and singular configurations. Effective computation procedures for a search for the minimum–maximum CSV are developed. The minimum CSV is applied to construct a PKM workspace satisfying preliminary given stiffness limits and compute a virtual protective barrier, keeping the PKM mechanism from approaching the vicinity of singular configurations. As an application example, the CSV-based local and global stiffness features of the Gough–Stewart platform mechanism are simulated, investigated, and visualized.

Stiffness analysis and experimental validation of robotic systems

Stiffness can be considered of primary importance in order to guarantee the successful use of any robotic system for a given task. Therefore, this paper proposes procedures for carrying out both numerical and experimental estimations of stiffness performance for multibody robotic systems. The proposed numerical procedure is based on models with lumped parameters for deriving the Cartesian stiffness matrix. Stiffness performance indices are also proposed for comparing stiffness performance. Then, an experimental procedure for the evaluation stiffness performance is proposed as based on a new measuring system named as Milli-CATRASYS (Milli Cassino Tracking System) and on a trilateration technique. Cases of study are reported to show the soundness and engineering feasibility of both the proposed numerical formulation for stiffness analysis and experimental validation of stiffness performance.

Joint stiffness identification of six-revolute industrial serial robots

Although robots tend to be as competitive as CNC machines for some operations, they are not yet widely used for machining operations. This may be due to the lack of certain technical information that is required for satisfactory machining operation. For instance, it is very difficult to get information about the stiffness of industrial robots from robot manufacturers. As a consequence, this paper introduces a robust and fast procedure that can be used to identify the joint stiffness values of any six-revolute serial robot. This procedure aims to evaluate joint stiffness values considering both translational and rotational displacements of the robot end-effector for a given applied wrench (force and torque). In this paper, the links of the robot are assumed to be much stiffer than its actuated joints. The robustness of the identification method and the sensitivity of the results to measurement errors and the number of experimental tests are also analyzed. Finally, the actual Cartesian stiffness matrix of the robot is obtained from the joint stiffness values and can be used for motion planning and to optimize machining operations.

Compound stiffness modelling of an integrated open-die forging centre with serial–parallel heavy-duty manipulators

This paper presents a method for modelling the compound stiffness of an integrated open-die forging centre that consists of a forging press as well as a manipulator that handles the workpiece. Open-die forging has considerable differences to general machining processes due to the complex plastic deformation effects created by consecutive forging strikes. The manipulator must comply with the movement of the workpiece during forging. The stiffness of the integrated system mainly comes from two sources: the compliance of the manipulator and the elastic deformation of the workpiece during forging. First, the stiffness matrix of the workpiece is derived using the theory of mechanics of materials. Then, the complete Cartesian stiffness matrix of the manipulator is developed by using the conservative congruence transformation method. Finally, the compound stiffness model is constructed by combining these two stiffness matrices. A numerical algorithm is developed that is able to simulate the compliance motion of a programmed open-die forging process. Two numerical examples are presented to validate the proposed modelling method and numerical algorithm.

On Cartesian stiffness matrices in rigid body dynamics: an energetic perspective

Several Cartesian stiffness matrices for a single rigid body subject to a conservative force field are developed in this paper. The treatment is based on energetic arguments and an Euler angle parameterization of the rotation of the rigid body is employed. Several new representations for the stiffness matrix are obtained and the relation to other works on Cartesian stiffness matrices and Hessians is illuminated. Additional details are presented with respect to determining the Cartesian stiffness matrix for a pair of rigid bodies, as well as for a system of rigid bodies constrained to a plane.

The stiffness matrix in elastically articulated rigid-body systems

Discussed in this paper is the Cartesian stiffness matrix, which recently has received special attention within the robotics research community. Stiffness is a fundamental concept in mechanics; its representation in mechanical systems whose potential energy is describable by a finite set of generalized coordinates takes the form of a square matrix that is known to be, moreover, symmetric and positive-definite or, at least, semi-definite. We attempt to elucidate in this paper the notion of “asymmetric stiffness matrices”. In doing so, we show that to qualify for a stiffness matrix, the matrix should be symmetric and either positive semi-definite or positive-definite. We derive the conditions under which a matrix mapping small-amplitude displacement screws into elastic wrenches fails to be symmetric. From the discussion, it should be apparent that the asymmetric matrix thus derived cannot be, properly speaking, a stiffness matrix. The concept is illustrated with an example.

Finding symmetric orthogonal Gough-Stewart platforms

This paper develops new, analytical methods to find a large class of orthogonal Gough-Stewart platforms (OGSPs) having desired properties at their home position. In contrast, prior methods have been computationally intensive, relying on numerical search techniques. By exploiting symmetry, 27 equations are reduced to only two. The new techniques are directly applicable to clean-sheet design of micro-manipulators, vibration isolators, and Cartesian stiffness matrices. In addition, straightforward methods for retro-fitting existing OGSPs are illustrated. Because the new theory greatly simplifies OGSP formulas about a single point, it is expected that these results will also prove to be very useful when numerically designing gross motion platforms

역나선 이론을 이용한 저자유도 병렬형 기구의 강성해석

This paper presents a methodology for the stiffness analysis of a low-DOF parallel manipulator. A low-DOF parallel manipulator is a spatial parallel manipulator which has less than six degrees of freedom. The reciprocal screws of actuations and constraints in each leg can be detemlined by making use of the theory of reciprocal screws, which provide information about reaction forces due to actuations and constraints. When pure force is applied to a leg, the leg stiffness is modeled as a linear spring along the line. For pure couple, it is modeled as a rotational spring about the axis. It is shown that the stiffness model of an F-DOF parallel nipulator consists of F springs related to actuations and 6-F springs related to constraints connected from the moving platform to the base in parallel. The 6×6 Cartesian stiffness matrix is derived, which is the sum of the Cartesian stiffness matrices of actuations and constraints. Finally, the 3-UPU, 3-PRRR, and Tricept parallel manipulators are used as examples to demonstrate the methodology.

A Geometrical Approach to the Study of the Cartesian Stiffness Matrix

The stiffness of a rigid body subject to conservative forces and moments is described by a tensor, whose components are best described by a 6×6 Cartesian stiffness matrix. We derive an expression that is independent of the parameterization of the motion of the rigid body using methods of differential geometry. The components of the tensor with respect to a basis of twists are given by evaluating the tensor on a pair of basis twists. We show that this tensor depends on the choice of an affine connection on the Lie group, SE3. In addition, we show that the definition of the Cartesian stiffness matrix used in the literature [1,2] implicitly assumes an asymmetric connection and this results in an asymmetric stiffness matrix in a general loaded configuration. We prove that by choosing a symmetric connection we always obtain a symmetric Cartesian stiffness matrix. Finally, we derive stiffness matrices for different connections and illustrate the calculations using numerical examples.

On the 6 × 6 cartesian stiffness matrix for three-dimensional motions

It has been recently observed by Pigoski et al. [1, 2], and by Ciblak and Lipkin [3] that the Cartesian stiffness matrix associated with a linear elastic coupling between two rigid bodies is, in general, asymmetric if the resulting forces and moments do not sum to zero. In this paper, methods of differential geometry and properties of Lie groups are used to show that in any conservative system subjected to a non-zero external load, if motions in SE(3), the special Euclidean group of rigid body motions in three dimensions, are considered, the resulting 6 × 6 Cartesian stiffness matrix is asymmetric in any inertial (fixed) or body-fixed reference frame. We prove the general result that in any subgroup of SE(3), if the system is subject to external forces and moments, the Cartesian stiffness matrix is symmetric if finite displacements along the basis twists are used to generate the stiffness matrix commute. Further, we derive several useful properties of stiffness matrices using ideas from Lie theory. In particular, we offer a simple proof to show that the stiffness matrix in the body-fixed reference frame is the transpose of the stiffness matrix in the inertial reference frame, a result also derived in [3]. Finally, we outline a method to construct a symmetric stiffness matrix by choosing an appropriate moving reference frame that is not fixed to any rigid body.

An efficient and locking-free flat anistropic plate and shell triangular element

The present study discusses, in a unified manner, the application of generalised decomposition principles and physical lumping techniques in finite element analysis of isotropic and laminated plates and shells. Using physical ideas and basic relations of mechanics, as well as geometrical arguments, it attempts to reduce the apparent complexity of the kinematical behaviour of such structures by means of assigning rigid-body and straining modes of deformation to an assembly of elements which comprise a compatible kinematical idealization. Subsequently, it applies these assertions to study a flat three-node 18 degrees-of-freedom triangular shell element applicable to the analysis of isotropic and fibre-reinforced laminated composite structures. Following the presentation of the general thermoelastic lamination theory, the Natural Mode Method is employed to form the elemental natural and Cartesian stiffness matrices. Although the elemental Cartesian stiffness matrix is of the order 18 × 18 and the initial Cartesian thermal load vector of the order 18 × 1, the triangular element essentially incorporates 12 generalized freedoms (the natural straining modes), and therefore neccesitates the computation of a 12 × 12 (not fully populated) natural stiffness matrix, and a 12 × 1 thermal (initial) vector, which to our knowledge make the present element the most inexpensive flat shell element available. The features of economy, simplicity and accuracy, facilitate the application of the present element in the non-linear range, in CPU intensive applications such as optimisation and failure — in general in the analysis of large and complex shell structures. The triangular finite element is supported by significant postprocessing features including the computation of all through-the-thickness stresses being equally important for structures composed from composite materials. Numerical examples are presented for both isotropic and laminated composite structures, and comparisons with analytic solutions are attempted where available. All computational experiments illustrate the accuracy of the conceived simple triangular element and substantiate its physical and theoretical bases.