Optimal Shape Control Via L_∞ Loss for Composite Fuselage Assembly

Shape control is critical to ensure the quality of composite fuselageassembly. In current practice, the structures are adjusted to the design shapein terms of the ℓ_2 loss for further assembly without considering theexisting dimensional gap between two structures. Such practice has twolimitations: (1) the design shape may not be the optimal shape in terms of apair of incoming fuselages with different incoming dimensions; (2) the maximumgap is the key concern during the fuselage assembly process. This paperproposes an optimal shape control methodology via the ℓ_∞ loss forcomposite fuselage assembly process by considering the existing dimensional gapbetween the incoming pair of fuselages. Besides, due to the limitation on thenumber of available actuators in practice, we face an important problem offinding the best locations for the actuators among many potential locations,which makes the problem a sparse estimation problem. We are the first to solvethe optimal shape control in fuselage assembly process using the ℓ_∞model under the framework of sparse estimation, where we use the ℓ_1penalty to control the sparsity of the resulting estimator. From statisticalpoint of view, this can be formulated as the ℓ_∞ loss based linearregression, and under some standard assumptions, such as the restrictedeigenvalue (RE) conditions, and the light tailed noise, the non-asymptoticestimation error of the ℓ_1 regularized ℓ_∞ linear model isderived to be the order of O(σ√(Slog p/n)), which meets theupper-bound in the existing literature. Compared to the current practice, thecase study shows that our proposed method significantly reduces the maximum gapbetween two fuselages after shape adjustments.