Flexible robot links challenge the rigid-body assumption, making joint measurements alone insufficient for accurate manipulation. This paper introduces a flexibility-aware manipulation framework that connects structural dynamics directly to sensing, kinematics, and control. Instead of prescribing a dense inertial measurement unit (IMU) network, the dominant vibration modes of each flexible link are first identified from response-convergence and excitation-bandwidth criteria. Under a full-rank modal observability condition, $m$ retained modes require only $m$ independent slope measurements and one reference IMU, defining the minimum observable IMU configuration. The resulting measurements are used to reconstruct the continuous link deformation, which is then incorporated into flexibility-aware forward kinematics and a deformation-informed partial differential equation-based controller. Experiments on hydraulic single- and double-link manipulators evaluate the framework. On the long-reach double-link platform, the proposed kinematics achieves end-effector root mean square errors of $0.0107$~m, $0.0194$~m, and $0.0102$~rad while using fewer sensors than the piecewise-rigid method. Closed-loop reaching experiments further demonstrate real-time integration of IMU-estimated joint states and reconstructed deformation into the proposed controller. The results show that dominant modal information can determine sensing requirements and enable real-time flexibility-aware robotic manipulation.

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