Motion coordination algorithms resulting from classical geometric optimization problems

2004 ◽  
Author(s):  
Jorge Cortés
Keyword(s):  
Optimization Problems ◽  
Geometric Optimization ◽  
Motion Coordination ◽  
Coordination Algorithms

Origami Folding and Bar Frameworks

2014 ◽  
Author(s):  
Alejandro R. Diaz
Keyword(s):  
Sensitivity Analysis ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Optimization Problems ◽  
Undirected Graph ◽  
Geometric Optimization ◽  
Order Effects ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Higher Order Effects

One of the more computationally demanding tasks in a process of synthesizing “from scratch” origami crease patterns designed for a given purpose involve a simulation capability to track the progression of the folding process as the pattern folds. This work presents an approach to simulate origami folding based on bar frameworks. The work is related to joint frameworks and projected polyhedral, as they apply to folding. The analysis starts from a representation of a crease pattern as an undirected graph G(E,V) formed by edges E and vertices V. A framework G(p) is an instance of G where the vertex locations are assigned positions according to a vector valued function p(t), where t marks the folding progression and t=0 represents the initial, flat configuration. The strategy presented is based on finding a sequence of instances {p(1), p(2), …} corresponding to an analytic flex p, i.e., functions such that edges in all G(p(t)) have the same length. The method is based on using a finite element description of a bar framework corresponding to a truss-like structure congruent with G(p). Solutions to an eigenvalue problem associated with this structure provide the means to update from p(t) to p(t+1). Two simple (purely geometric) optimization problems adjust the update to compensate for higher order effects, guaranteeing that the length of the edges remain constant. The methodology can be used to achieve configurations close to “flat folding”, provided that no interference of the faces occurs along the way. We expected that physically-motivated constraints (stresses, deformations, etc.) and sensitivity analysis computations will be more easily represented in this framework and therefore this formulation will have an advantage over more standard “origami mathematics” approaches. The approach is illustrated with an example of folding a simple 10-crease pattern.

scholarly journals On some geometric optimization problems in layered manufacturing

1999 ◽  
Vol 12 (3-4) ◽  
pp. 219-239 ◽  
Author(s):  
Jayanth Majhi ◽  
Ravi Janardan ◽  
Michiel Smid ◽  
Prosenjit Gupta
Keyword(s):  
Optimization Problems ◽  
Layered Manufacturing ◽  
Geometric Optimization

scholarly journals Constrained Nets for Graph Matching and Other Quadratic Assignment Problems

1991 ◽  
Vol 3 (2) ◽  
pp. 268-281 ◽  
Author(s):  
Petar D. Simić
Keyword(s):  
Graph Matching ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Semiclassical Approximation ◽  
Geometric Optimization ◽  
Elastic Net ◽  
Physical Models ◽  
Quadratic Assignment ◽  
Matching Problem ◽  
Special Case

Some time ago Durbin and Willshaw proposed an interesting parallel algorithm (the “elastic net”) for approximately solving some geometric optimization problems, such as the Traveling Salesman Problem. Recently it has been shown that their algorithm is related to neural networks of Hopfield and Tank, and that they both can be understood as the semiclassical approximation to statistical mechanics of related physical models. The main point of the elastic net algorithm is seen to be in the way one deals with the constraints when evaluating the effective cost function (free energy in the thermodynamic analogy), and not in its geometric foundation emphasized originally by Durbin and Willshaw. As a consequence, the elastic net algorithm is a special case of the more general physically based computations and can be generalized to a large class of nongeometric problems. In this paper we further elaborate on this observation, and generalize the elastic net to the quadratic assignment problem. We work out in detail its special case, the graph matching problem, because it is an important problem with many applications in computational vision and neural modeling. Simulation results on random graphs, and on structured (hand-designed) graphs of moderate size (20-100 nodes) are discussed.

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