Goizalde Ajuria

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.

A finite element approach to the position problems in open-loop variable geometry trusses

The present paper looks at some kinematic and static-equilibrium problems that arise with variable-geometry trusses (VGTs). The first part of the paper looks at the use of active controls in the correction of static deformations, the second part at the position problems. The separation between deformable- and rigid-body displacements makes it possible to consider separately the corrections in each component of the structure. VGTs are considered as open-loop linkages with redundant rigid-body degrees of freedom. Owing to this redundancy, possible solutions to the inverse problem are in general infinite, for which reason it is necessary to use some optimization criteria. To tackle the problem an optimization procedure with constraints is developed for the purpose of minimizing the displacements of the actuators. Suitable use of the constraints allows us to solve the direct position problem using the same optimization procedure.

Comparison among nonlinear optimization methods for the static equilibrium analysis of multibody systems with rigid and elastic elements

Abstract The present paper describes a set of procedures for the solution of the nonlinear static-equilibrium problem in complex multibody mechanical systems, including rigid and elastic elements. The error function is a simple one based on the potential function, which includes rigid elements by means of nonlinear constraints. To this end Lagrange Multipliers, along with various versions of the Augmented Lagrange Multipliers (ALM) or Primal–Dual Method are used, comparisons are made between them. A Newton–Raphson second-order method is used in seeking function minima for equilibrium positions. This procedure is also directly useful for other mechanism position problems, such as initial position, finite displacements, and deformed position, as well as direct and inverse problems.

Lagrange multipliers and the primal–dual method in the non‐linear static equilibrium of multibody systems

This paper presents four different approaches to the solution of the non-linear static-equilibrium problem in complex linkages, including rigid and elastic elements. The error function is based on the potential of the system, and includes rigid elements by means of non-linear constraints. To this end use is made of Lagrange multipliers, along with the primal-dual method, penalty functions and weighted stiffness, comparisons being made between them. A Newton-Raphson method is used in seeking function minima for equilibrium positions. This procedure is also directly applicable to the other linkage and multibody position problems.

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.