Coordinated arm-hand motion planning is fundamental to dexterous robotic manipulation in complex and constrained environments. A straightforward solution is to decompose the problem into separate arm path planning and hand motion generation; however, this poses a dilemma: decomposition can miss feasible solutions that require coordinated arm-hand adaptation along the path. Alternatively, directly planning in the high-dimensional joint arm-hand configuration space captures such coupling but faces a substantially enlarged search space and nonconvex collision constraints. To characterize this coupling, we formulate feasible hand fibers that capture collision-free hand configurations for each arm configuration. Based on this formulation, we propose CAMP, a high-success and efficient cooperative arm-hand motion planner for constrained environments. CAMP constructs candidate trajectories through layered hand search with local arm relaxation, then compactly represents them using endpoint-preserving via-point movement primitives (VMPs) for coarse-to-fine joint optimization. Across six constrained simulation tasks, CAMP achieves 84.2-98.5% planning success, outperforming alternative planners with competitive efficiency. Ablation studies verify the contributions of arm relaxation, VMP representation, and coarse-to-fine optimization, while real-robot experiments demonstrate CAMP on constrained manipulation tasks. The project website is available at https://camp-armhand.github.io/.

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