Dexterous grasping with multi-fingered hands remains challenging due to high-dimensional articulations and the cost of optimization-based pipelines. Existing end-to-end methods require training on large-scale datasets for specific hands, limiting their ability to generalize across different embodiments. We propose an eigengrasp-based, end-to-end framework for cross-embodiment grasp generation. From a hand's morphology description, we derive a morphology embedding and an eigengrasp set. Conditioned on these, together with the object point cloud and wrist pose, an amplitude predictor regresses articulation coefficients in a low-dimensional space, which are decoded into full joint articulations. Articulation learning is supervised with a Kinematic-Aware Articulation Loss (KAL) that emphasizes fingertip-relevant motions and injects morphology-specific structure. In simulation on unseen objects across three dexterous hands, our model attains a 91.9% average grasp success rate with less than 0.4 seconds inference per grasp. With few-shot adaptation to an unseen hand, it achieves 85.6% success on unseen objects in simulation, and real-world experiments on this few-shot generalized hand achieve an 87% success rate. The code and additional materials will be made available upon publication on our project websitethis https URL.
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