Large telescopes present a great design challenge for minimizing the material used while also satisfying severe shape and other performance constraints. Using less mass and material resources will be important for designers of the next generation of structures trying to squeeze better performance out of ever more scarce natural resources. But mass also plays an important role in the shape control of telescopes. The larger the mass, the larger are the gravitational forces that distort the shape. As the telescope changes its position (elevation angle), the gravity forces change. Hence, to make a traditional truss structure stiff enough to maintain optical performance requires a very massive structure.

In our approach,^{1–5} we seek the smallest mass that can tolerate the largest gravity forces, while adjusting the tensions in certain members to keep the shape parabolic within optical standards in every telescope position. The structural paradigm that provides the smallest mass is a very specialized kind of truss called tensegrity.^{6}

A tensegrity structure is a network of axially loaded bars and cables that are stabilizable by prestress. Thus, even though the overall structure bends, no individual member bends. The absence of material bending (no clamps, rivets, or welding are required) is the key to great mass reduction. Our motivation comes from biology,^{6} but the term tensegrity comes from Buckminster Fuller to describe an art form.

Our recent work solved five fundamental problems in engineering mechanics, and the solutions to all five problems were tensegrity types of structures.^{1–5} We found explicit analytical formulas for these five different structure optimization problems that were traditionally reachable only by large, inaccurate computations. These optimization problems minimized mass subject to yield constraints for five different boundary conditions (cantilever bending, simply-supported bending, compressive loads, tensile loads, and torsional loads). Since any loaded structure (e.g., bridge, building, telescope) has components that are exposed to one or more of these types of loads (bending, compression, or tension), these analytical results provide insight into the efficient design of more complex structures.

The second contribution of these results^{1} is the notion of optimal complexity. For each of the five fundamental problems cited above, there is an optimal complexity, *q* (the number of elements in the final structure), associated with each design. If one uses more or fewer than *q* cables and bars, the required mass will be larger. The number *q* of components, which have to be connected, also relates to construction costs, providing an additional insight for the design.

We have designed three systems whose mass savings have been calculated: a footbridge (for architects in Switzerland) that has one third the mass of traditional designs; a wing design that is 30% lighter than the best existing wings; and a parabolic telescope design that is less than one fifth the mass of a previous traditional truss design.

**Figure 1. **Parabolic tensegrity rigid structure.

Figure 1 shows our latest parabolic tensegrity telescope concept. We believe this is the first tensegrity structure (other than a simple unit^{1}) that has no soft modes (i.e., it is stiff in all directions). By adjusting the tension of certain prestressed cables, one can maintain a parabolic shape for all positions of the telescope. We are now deriving the optimal complexity for this minimal-mass telescope.

The control of cable tensions requires feedback information such as force transducers in certain tension and compressive components of the structure. The design of the control algorithm should minimize the energy that would be required to suppress vibrations below a specified level and to keep the parabolic shape errors below optical tolerances. The grand challenge before us is to design a structure with minimal mass that also requires the smallest control energy to meet the performance requirements. This will be among the first attempts to truly integrate the design of structure and control rather than the traditional approach of designing the control only after the structure is determined. This integration of the control and structure design disciplines will certainly be required to meet the more stringent demands of future design challenges. Achieving this integrated approach is the focus of our ongoing research.

Robert Skelton

University of California, San Diego

La Jolla, CA

Robert Skelton is professor emeritus of the Mechanical and Aerospace Engineering Department. He is a member of the National Academy of Engineering, a Fellow of the American Institute of Aeronautics and Astronautics, and a Fellow of the Institute of Electrical and Electronic Engineers. He was co-recipient of the Nichols Medal of the American Society of Civil Engineers, and won international research awards from Germany's Alexander von Humboldt Foundation and the Japanese Society for the Promotion of Science. He served on the National Research Council and the Aeronautics and Astronautics Engineering Board. He developed control systems for both Skylab and the Hubble Space Telescope, and served on the External Independent Readiness Review (EIRR) panel for Hubble repair missions. He has published five books and 250 journal papers.

References:

1. R. E. Skelton, M. C. de Oliveira, *Tensegrity Systems* , Springer, New York, 2009.

2. R. E. Skelton, M. C. de Oliveira, Optimal complexity of deployable compressive structures, *J. Franklin Institute* 347(1), p. 228-256, 2010.

3. R. E. Skelton, M. C. de Oliveira, Optimal tensegrity structures in bending: the discrete Michell truss, *J. Franklin Institute* 347(1), p. 257-283, 2010.

4. K. Nagase, R. Skelton, Tensegrity design by convex optimization, , 2013. To appear. Available now as University of California, San Diego, internal report

5. K. Nagase, R. Skelton, Tensegrity systems with a common connectivity, , 2013. To appear. Available now as University of California, San Diego, internal report.

6. K. Snelson, Snelson on the Tensegrity Invention, *Int. J. Space Struct.* 11, p 43, 1996.