Finite element method with local damage of the mesh

We consider the finite element method on locally damaged meshes allowing for some distorted cells which are isolated from one another. In the case of the Poisson equation and piecewise linear Lagrange finite elements, we show that the usual a priori error estimates remain valid on such meshes. We also propose an alternative finite element scheme which is optimally convergent and, moreover, well conditioned, i.e. the conditioning number of the associated finite element matrix is of the same order as that of a standard finite element method on a regular mesh of comparable size.

Dimensionally Homogeneous Jacobian and Condition Number

This paper deals with the formulation of the dimensionally homogeneous extended Jacobian matrix, which is an important issue for the performance analysis of f degrees-of-freedom (f ≤6) parallel manipulators having coupled rotational and translational motions. By using the f independent coordinates to define the permitted motions and (6-f) independent coordinates to define the restricted motions of the moving platform, the 6×6 dimensionally homogeneous extended Jacobian matrix is derived for non-redundant parallel manipulators. The conditioning number of the parallel manipulators is computed to evaluate the homogeneous extended Jacobian matrix, the homogeneous actuation wrench matrix, and the homogeneous constraint wrench matrix to evaluate the performance of the parallel manipulators. By using these indices, the closeness of a pose to different singularities can be detected. An illustrative example with the 3-RPS parallel manipulator is provided to highlight the effectiveness of the approach and the proposed indices.

Dimensional Synthesis of a Planar Parallel Mechanism for Wafer Transfer with Kinematics Performance

This paper deals with the dimensional synthesis of a planar parallel mechanism for wafer transfer with kinematics performance. Under the relationship between workspace and parametric, a kinematics performance index, which is composed of the conditioning number of the Jacobian matrix and the motion resolution, is considered as the evaluation criterion. And a new index which is proposed by considering both the mean value and fluctuation of the pre-mentioned index is used to optimally design the dimensional parameters. It is testified that the optimal dimensional parameters can satisfied with the kinematics performance.

Sensitivity Analysis of Planar Parallel Manipulators

This paper deals with the sensitivity analysis of planar parallel manipulators. A methodology is introduced to derive the sensitivity coefficients by means of the study of 3-RPR manipulators. As a matter of fact, the sensitivity coefficients of the pose of its moving platform to variations in the geometric parameters are expressed algebraically, the variations being defined both in Polar and Cartesian coordinates. The dexterity of the manipulator is also studied by means of the conditioning number of its normalized kinematic Jacobian matrix. As an illustrative example, the sensitivity of a symmetrical planar parallel manipulator is analyzed in detail. Finally, the accuracy of the manipulator is compared with its dexterity.

RATING THE PERFORMANCE OF SHOOTING METHODS FOR THE COMPUTATION OF PERIODIC ORBITS

By means of a conditioning number for linear boundary value problems (BVP) and a stability measure for shooting methods a strategy is outlined to rate the performance of shooting methods for the calculation of periodic orbits in dynamical systems. With help of this strategy two example models are rated, with an astonishing result: the single shooting has got a better performance. In both cases it will be shown that the choice of phase condition has no influence on the performance.

Optimal Kinematic Design of Gantry Parallel Robots

This paper addresses some issues of optimal kinematic design of parallel robot with 3 degrees of freedom. Robot performs translational gantry type motion with a large workspace. The structure’s nominal kinematics is identical to the Linear Delta Robot, the difference being the way which the links are organized. In order to yield a design of the Gantry Parallel Robot with a large workspace volume and good kinematic properties, multi-criteria optimization is used. The objective functions comprise a space utilization index and the inverse conditioning number of the Jacobian matrix. Three different geometrical layouts of the robot are examined and the Pareto optimality between the objective functions is found for several different cases. Pareto-optimal curves showing the relationship between the workspace size and the manipulability are presented as a tool for the designer. The obtained results give a good overview of the kinematic properties of the Gantry Parallel Robot that can be used for designing the parallel structures for various applications.

Controllability Index Based on Conditioning Number

Use of the conditioning number k(F) = ∥ F ∥ • ∥ F−1 ∥ where F is a symmetric matrix related to the controllability (observability) matrix is suggested as basis for index of controllability (observability). This index has advantages of being independent of scaling of state variables and being a continuous function of parameters of process.