Composite structures in aeroplanes are often thin-walled and lightweight, resulting in significant compliance, which presents a handling and assembly challenge due to the associated part deformations. In order to counteract these deformations, the parts are held in their specified geometry using stiff and correspondingly heavy fixtures or jigs. Mobile industrial robots are very versatile and widely used in industrial volume production, but they are limited in their payload capacity. High-rate production of large aerospace modules requires highly automated flexible assembly processes. The approach presented in this paper is to combine mobile units with lightweight assembly jigs that have the capability of deformation compensation. The subject of the study is a high-rate assembly process for flap modules using an Autonomous Industrial Mobile Manipulator (AIMM) and a lightweight end effector. The end effector has a shape compensation function, implemented by an integrated Stewart platform, which enables the compensation of manufacturing tolerances as well as gravity effects. The compensation function is used in a closed loop and counteracts shape deviations by appropriate fixture shape adjustments. The paper reports on the conceptual design of the assembly scenario, the design of the end effector, its realization and the successful experimental demonstration at 1:1 scale.