The solution of variable-geometry truss problems using new homotopy continuation methods
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A VGT or Variable-Geometry Truss can be thought of as a statically determinate truss that has been modified to contain some number of variable length members. These extensible members allow the truss to vary its configuration in a controlled manner. VGTs are often symmetric. constructed of repeating cells, and have exceptional stiffness to weight ratios. VGTs have been widely recognized as adaptive or collapsing structures for space and military applications. Some of the typical applications envisioned for VGTs are as booms to position equipment in space. as supports for space antennae. and as berthing devices. Lately, they have also been proposed as parallel-actuated. long chain. high dexterity manipulators. This work describes basic VGT theory. and presents criteria to use in the determination of valid VGT unit cells. Four of the VGT unit cells - the tetrahedron. the octahedron, the decahedron. and the dodecahedron are discussed in detail. The typical modeling and formulation procedure for developing the kinematic equations associated with the forward kinematic problem of each of the above is described. Another intent of this work is to present a new and efficient technique for solving the forward kinematics problem of VGTs. All VGT problems lead to systems of equations. Commonly, such systems are solved by an iterative numerical method, usually a Newton method or a variant. For such methods to yield a solution, a starting point sufficiently close to the actual solution must be supplied. For systems of the size of those encountered in VGT problems, this is a formidable task. On the other hand, recently developed methods in homotopy continuation for polynomials are not only global, but also exhaustive; i.e., they do not require good initial guesses and they also guarantee convergence to all solutions. Homotopies are a traditional part of topology and have only recently begun to be used for practical numerical computation. Polynomial continuation is used to track the solutions of the systems of equations describing the kinematics of VGTs. This method has proven to be robust and reliable. It may also prove to be a valuable tool in the analysis of other kinematic devices with a high multiplicity of solutions.
- Doctoral Dissertations