Design, Optimization, and Control of Tensegrity Structures

✍ Scribed by Milenko Masic

• Publisher: ProQuest Dissertations And Theses
• Year: 2004
• Tongue: English
• Leaves: 172
• Series: Dissertation Abstracts International B 65-03
• Edition: 1
• Category: Library

✦ Synopsis

ABSTRACT OF THE DISSERTATION
Design, Optimization and Control of Tensegrity Structures by Milenko Masic
Doctor of Philosophy in Engineering Sciences (Aerospace Engineering)
University of California , San Diego, 2004
Professor Robert E. Skelton , Chair

The contributions of this dissertation may be divided int o four categories.

The first category involves developing a systematic form-finding method for general and symmetric tensegrity structures. As an extension of the available results, different shape constraints are incorporated in the problem. Methods for treatment of these constraints are considered and proposed. A systematic formulation of the form-finding problem for symmetric tensegrity structures is introduced , and it uses the symmetry to reduce both the numb er of equations and the number of variables in the problem. The equilibrium analysis of modular tensegrities exploits their peculiar symmetry. The tensegrity similarity transformation completes the contributions in the area of enabling tools for tensegrity form-finding.

The second group of contributions develops the methods for optima l mass-to-stiffness-ratio design of tensegrity structures. This technique represents the state-of-the-art for the static design of tensegrity structures. It is an extension of the results available for the topology optimization of truss structures. Besides guaranteeing that the final design satisfies the tensegrity paradigm , the problem constrains the structure from different modes of failure, which makes it very general.

The open-loop control of the shape of modular tensegrities is the third contribution of the dissertation. This analytical result offers a closed form solution for the control of the reconfiguration of modular structures. Applications range from the deployment and stowing of large-scale space structures to the locomotion-inducing control for biologically inspired structures. The control algorithm is applicable regardless of the size of the structures, and it represents a very general result for a large class of tensegrities. Controlled deployments of large-scale tensegrity plates and tensegrity towers are shown as examples that demonstrate the full potential of this reconfiguration strategy.

The last contribution of the dissertation represents the method for integrated structure and control design of modular tensegrity structures. A gradient optimization method is used for this particular class of problems, and it proves to be very efficient. The examples that are given demonstrate the impact of the distribution of the prestress on the optimal dynamic performance of the structure.

✦ Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 History of tensegrity structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation for the research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Summary of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Statics of tensegrity structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 The tensegrity equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Shape constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Tensegrity structure stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Tensegrity form-finding examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Enabling tools for tensegrity form-finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Invariant tensegrity geometric transformations . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Composition of tensegrity structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Geometry and equilibrium analysis of some tensegrity modules . . . . . . . . . . . . . . . . . . 53
3.5 Geometry and equilibrium of monohedral modular tensegrity plates . . . . . . . . . . . . . . . . 68
3.6 Geometry and equilibrium of class-two tensegrity towers . . . . . . . . . . . . . . . . . . . . .77
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78
4 Open-loop control of modular tensegrities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Slowly varying nonlinear systems and open-loop control . . . . . . . . . . . . . . . . . . . . . 81
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88
5 Optimal mass-to-stiffness-ratio tensegrity design . . . . . . . . . . . . . . . . . . . . . . . . .90
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 Nonlinear program formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100
5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113
6 Joint structure and control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Lumped mass dynamic model of a tensegrity structure . . . . . . . . . . . . . . . . . . . . . . .116
6.3 Linearized dynamic model of the structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
6.4 Designing the structure for the optimal LQR performance - optimization over the prestress cone . 125
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136
7 General conclusions and future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137
7.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138
8 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.A Equivalency of the force density and length-minimization method for tensegrity form-finding . . .140
8.B Symmetry of prestress forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.C Open-loop control laws for typical elements of modular tensegrities . . . . . . . . . . . . . . .148
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

✦ Subjects

Applied sciences; Mass-to-stiffness; Tensegrity; Aerospace materials; 0538:Aerospace materials