The surface accuracy of an inflatable rectangular membrane subjected to pressure and isotropic tension in a reflector is investigated. The inflatable membrane is assumed to be placed at an arbitrary location in a reflector with an arbitrary rotation angle alpha between the side of the membrane and the principal direction on the ideal parabolic surface. The edges of the membrane are assumed to coincide with an elliptic or a hyperbolic paraboloidal surface. The deflection of the membrane is obtained analytically based on the linear equation (Poissons Equation) for a pressurized membrane. Different kinds of surface error optimizations with respect to the ratio of pressure and tension q and parameters that determine the edge deflection of the membrane are carried out. The effects of q, alpha, the aspect ratio and the location g of the membrane upon each surface error of the optimum membranes are examined. The surface error of an inflatable rectangular membrane whose edges coincide with the approximate parabolic surface (APS) of a reflector is found to be zero on the assumption that the appropriate ratio of pressure and tension is applied. With the exception of an inflatable flat-edge-optimum membrane, each surface error of inflatable optimum membranes takes a value of zero or increases monotonically as the distance from the vertex increases, and the surface error also takes a value of zero or constant, or decreases as the aspect ratio increases or decreases away from one. Moreover the surface error is found to be well below that of the unpressurized optimum membranes when the membranes are located near the vertex of the reflector.

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This paper proposes optimization of the PD controller gains by minimizing H infinity norm for flexible spacecraft attitude control. Solving the problem is known to be difficult because it is described by a bilinear matrix inequality (BMI) that cannot be transformed into linear matrix inequalities (LMIs). In order to solve it within practical computation time, we revise the branch and bound algorithm and incorporate the quantization of the variable domains. The ETS-VI mathematical model is used to demonstrate its ability.

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The surface accuracy of a rectangular membrane subjected to anisotropic tension in a reflector surface is investigated. The rectangular membrane is assumed to be located at an arbitrary point on the reflector surface with an arbitrary rotation angle alpha between the side of the membrane and the direction of the principal curvature of the reflector surface. The deflection and surface error of the membrane are calculated based on an analytical solution of the linear membrane equation. Three kinds of rectangular optimum membranes, whose surface errors are minimized with respect to parameters that specify the edge deflections, are considered and evaluated. These surface errors, D(1) (surface error of an equal-curvature-edge-optimum membrane), D(2opt) (surface error of an optimum-2 membrane), D(3) (surface error of an optimum-3 membrane), are also compared with D(copt) (surface error of a coincident-optimum-edge membrane) that is studied in other literature. In a certain range of parameters, it is found that the appropriate tension ratio reduces the surface error of each optimized membrane as compared to isotropic tension. The more slender the membrane is and the higher the ratio of tension (ratio of higher tension on the longer side to lower tension on the shorter side) is, the smaller the surface error becomes. The minimum surface error is obtained when the longer side is placed parallel to the direction of the larger principal curvature at the location and the longer side is loaded with higher tension. When alpha=0, D(2opt) is equal to D(3). As for a slender membrane whose longer side is loaded with higher tension, D(2opt) [equal to D(3)] is approximately equal to D(1) and D(copt) when alpha is equal to 0, and D(2opt) [approximately equal to D(copt)] is smaller than D(1) [approximately equal to D(3)] when alpha is not equal to 0. It is found that D(2opt) is the smallest among other membranes, and the surface error is about 0 - 15% less than D(copt). Each surface error decreases as the distance from the vertex increases.

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The surface accuracy of a rectangular membrane subjected to pressure and non-uniform tension in a reflector is investigated. The rectangular membrane is assumed to be located at an arbitrary point on the reflector surface with an arbitrary rotation angle between the side of the membrane and the direction of the principal curvature of the reflector surface. Rectangular membranes with two kinds of edge deflections are considered. One is a coincident-edge membrane, in which the edges coincide with the approximate parabolic surface at the local point where the membrane is placed. The other membrane is a coincident-edge-optimum membrane, which is obtained by minimizing the rms error using a normal translation of the coincident-edge membrane. The deflection and surface error of the membrane are calculated based on an analytical solution of the linear membrane equation. The effects of pressure, tension ratio parameter, rotation angle, aspect ratio and location of the membrane upon each membrane surface error are examined. It is found that the appropriate values of pressure and tension ratio parameter give the membrane zero surface error with arbitrary values of aspect ratio, rotation angle and location parameter. The condition for zero surface error does not depend on the aspect ratio. When the surface error is not zero, the ratio of the surface error of a coincident-edge-optimum membrane to that of a coincident-edge membrane is approximately 0.4 - 0.5 and depends only on the ratio of aspect ratio to tension ratio parameter.

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The surface accuracy of a N-sided regular polygonal membrane in a reflector is investigated analytically, where N is an integer of 3, 4, 5, 6, ---. The membrane is assumed to be located at an arbitrary point on the reflector surface with an arbitrary rotation angle between the side of the membrane and the direction of the principal curvature of the ideal parabolic surface. Membranes with two kinds of edge deflections are considered. One is a coincident-edge membrane whose edges coincide with the approximate parabolic surface of the local point where the membrane is placed. The second is a coincident-edge-optimum membrane, which is obtained by minimizing the rms error with the normal translation of the coincident-edge membrane. Analytical formulation for the deflection and rms error is obtained based on a linear membrane equation. Effects of edge deflections, number of sides N, location of the membrane, etc. upon the surface accuracy are examined. The surface error of the coincident-edge-optimum membrane is found to be about half that of the coincident-edge membrane. The surface error of each membrane decreases as the distance from the center axis of the ideal parabolic surface becomes larger and is not dependent on the rotation angle. The surface errors of coincident-edge and coincident-edge-optimum membranes increase monotonically as N increases. These surface errors converge to those of a circular membrane as N increases to infinity.

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