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This study presents the design and the kinematic optimization of a novel, underactuated, linkage-based robotic hand exoskeleton to assist users in performing grasping tasks. The device has been designed to apply only normal forces to the finger phalanges during flexion/extension of the fingers, while providing automatic adaptability for different finger sizes. Thus, the easiness of the attachment to the user's fingers and better comfort have been ensured. The analyses of the device kinematic pose, statics, and stability of grasp have been performed. These analyses have been used to optimize the link lengths of the mechanism, ensuring that a reasonable range of motion is satisfied while maximizing the force transmission on the finger joints. Finally, the usability of a prototype with multiple fingers has been tested during grasping tasks with different objects.