Multiple pneumatic artificial muscles (PAMs) connected through antagonistic joints are more in line with the motion characteristics of human muscles, which better imitate/replace humans to complete a series of actual tasks, such as transportation and assembly. However, there is still a lack of comprehensive solutions to handle hysteresis, creep, input delay, and other inherent characteristics of PAMs, as well as synchronous control and obstacle avoidance that are important to multiple muscles working together. To this end, this paper proposes a new neuroadaptive synchronization controller for 3-D antagonistic PAM-actuated robot hands, which also elaborately designs auxiliary terms to realize obstacle avoidance in Cartesian space and motion constraints in joint space together. Here, dynamic obstacles are regarded as external independent objects, whose nonlinear dynamics are introduced into the proposed controller to restrict end-effectors. Meanwhile, the constraint terms of joint angles and angle velocities are designed as time-varying proportional-differential gains, instead of common barrier functions that may induce overlarge inputs. Particularly, this paper proposes an accelerated gradient-based learning term to relax the linear parameterization condition of uncertain/unmodeled dynamics and obtain accurate weight estimates, based on which, it is proven that both tracking errors and synchronous errors rapidly converge to zero. In addition to complete theoretical analysis, some hardware experiments also verify the effectiveness and adaptability of the proposed controller.