J. Agirrebeitia

A method for the study of position in highly redundant multibody systems in environments with obstacles

This paper looks at a method for the analysis of highly redundant multibody systems (e.g., in the case of cellular adaptive structures of variable geometry) in environments with obstacles. Our aim is to solve the inverse kinematics in successive positions of multibody systems, avoiding the obstacles in its work environment. The multibody systems are modeled via rod-type finite elements, both deformable and indeformable, and the coordinates of their nodes are chosen as variables. The obstacles are modeled via a mesh of points that exert repulsive forces on the nodes of the model of the multibody, in order to model the obstacle avoidance. Such forces have been chosen inversely proportional to the Nth power of the distance between the corresponding points of the obstacle and of the multibody system. The method is based on a potential function and on its minimization using the Lagrange Multiplier Method. The solution of the resulting equations is undertaken iteratively with the Newton-Raphson Method.

Inverse position problem in highly redundant multibody systems in environments with obstacles

This paper looks at a method for the analysis of highly redundant multibody systems (e.g. in the case of cellular adaptive structures of variable geometry) in environments with obstacles. It is sought to solve the inverse problem in successive positions of multibody systems, avoiding the obstacles in its work environment; i.e. the computation of the increment that has to be assigned to the actuators throughout the movement of the multibody system so that it does not collide with obstacles, as one or more nodes perform a pre-established function (e.g. a certain path). The multibody systems are modelled via rod-type finite elements, both deformable and rigid, and the coordinates of their nodes are chosen as variables. The obstacles are modelled via a mesh of points that exert repulsive forces on the nodes of the model of the multibody, so that interference between the two is avoided. Such forces have been chosen inversely proportional to the Nth power of the distance between the corresponding points of the obstacle and of the multibody system. The method is based on a potential function and on its minimization using the Lagrange Multiplier Method. The solution of the resulting equations is undertaken iteratively with the Newton–Raphson method. The 2D and 3D examples provided attest to the good performance of the algorithms and procedure here set forth.

An alternative full-pivoting algorithm for the factorization of indefinite symmetric matrices

This paper presents an algorithm for the factorization of indefinite symmetric matrices that factors any symmetric matrix A into the form LDL^T, with D diagonal and L triangular, with its subdiagonal filled with zeros. The algorithm is based on Jacobi rotations, as opposed to the widely used permutation methods (Aasen, Bunch-Parlett, and Bunch-Kaufman). The method introduces little increase in computational cost and provides a bound on the elements of the reduced matrices of order 2nf(n), which is smaller than that of the Bunch-Parlett method (~3nf(n)), and similar to that of Gaussian elimination with full pivoting (nf(n)). Furthermore, the factorization method is not blocked. Although the method presented is formulated in a full-pivoting scheme, it can easily be adapted to a scheme similar to that of the Bunch-Kaufman approach. A backward error analysis is also presented, showing that the elements of the error matrix can be bounded in terms of the elements of the reduced matrices.

A Numerical Method for the Second-Order Mobility Analysis of Mechanisms

The second order mobility analysis of mechanisms is a complicated problem that can be approached in a direct way via the analysis of the compatibility of the acceleration field. This paper will present a simple, numerical approach to retrieve the restrictions imposed to the movement that are derived from the second order (curvature) restrictions. This algorithm can be easily applied to bi and three dimensional mechanisms and delivers a good degree of efficiency. The results of this analysis can be employed to improve the efficiency of other algorithms which present lack of convergence in the vicinity of singular configurations.