Optimal Shape Control via $L_\infty$ Loss for Composite Fuselage Assembly

Shape control is critical to ensure the quality of composite fuselage assembly. In current practice, the structures are adjusted to the design shape in terms of the $\ell_2$ loss for further assembly without considering the existing dimensional gap between two structures. Such practice has two limitations: (1) the design shape may not be the optimal shape in terms of a pair of incoming fuselages with different incoming dimensions; (2) the maximum gap is the key concern during the fuselage assembly process. This paper proposes an optimal shape control methodology via the $\ell_\infty$ loss for composite fuselage assembly process by considering the existing dimensional gap between the incoming pair of fuselages. Besides, due to the limitation on the number of available actuators in practice, we face an important problem of finding the best locations for the actuators among many potential locations, which makes the problem a sparse estimation problem. We are the first to solve the optimal shape control in fuselage assembly process using the $\ell_\infty$ model under the framework of sparse estimation, where we use the $\ell_1$ penalty to control the sparsity of the resulting estimator. From statistical point of view, this can be formulated as the $\ell_\infty$ loss based linear regression, and under some standard assumptions, such as the restricted eigenvalue (RE) conditions, and the light tailed noise, the non-asymptotic estimation error of the $\ell_1$ regularized $\ell_\infty$ linear model is derived to be the order of $O(\sigma\sqrt{\frac{S\log p}{n}})$, which meets the upper-bound in the existing literature. Compared to the current practice, the case study shows that our proposed method significantly reduces the maximum gap between two fuselages after shape adjustments.