Grasping is a fundamental skill for robots to interact with their environment. While grasp execution requires coordinated movement of the hand and arm to achieve a collision-free and secure grip, many grasp synthesis studies address arm and hand motion planning independently, leading to potentially unreachable grasps in practical settings. The challenge of determining integrated arm-hand configurations arises from its computational complexity and high-dimensional nature. We address this challenge by presenting a novel differentiable robot neural distance function. Our approach excels in capturing intricate geometry across various joint configurations while preserving differentiability. This innovative representation proves instrumental in efficiently addressing downstream tasks with stringent contact constraints. Leveraging this, we introduce an adaptive grasp synthesis framework that exploits the full potential of the unified arm-hand system for diverse grasping tasks. Our neural joint space distance function achieves an 84.7% error reduction compared to baseline methods. We validated our approaches on a unified robotic arm-hand system that consists of a 7-DoF robot arm and a 16-DoF multi-fingered robotic hand. Results demonstrate that our approach empowers this high-DoF system to generate and execute various arm-hand grasp configurations that adapt to the size of the target objects while ensuring whole-body movements to be collision-free.

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