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Beyond providing accurate movements, achieving smooth motion trajectories is a long-standing goal of robotics control theory for arms aiming to replicate natural human movements. Drawing inspiration from biological agents, whose reaching control networks effortlessly give rise to smooth and precise movements, can simplify these control objectives for robot arms. Neuromorphic processors, which mimic the brain's computational principles, are an ideal platform to approximate the accuracy and smoothness of biological controllers while maximizing their energy efficiency and robustness. However, the incompatibility of conventional control methods with neuromorphic hardware limits the computational efficiency and explainability of their existing adaptations. In contrast, the neuronal subnetworks underlying smooth and accurate reaching movements are effective, minimal, and inherently compatible with neuromorphic hardware. In this work, we emulate these networks with a biologically realistic spiking neural network for motor control on neuromorphic hardware. The proposed controller incorporates experimentally-identified short-term synaptic plasticity and specialized neurons that regulate sensory feedback gain to provide smooth and accurate joint control across a wide motion range. Concurrently, it preserves the minimal complexity of its biological counterpart and is directly deployable on Intel's neuromorphic processor. Using the joint controller as a building block and inspired by joint coordination in human arms, we scaled up this approach to control real-world robot arms. The trajectories and smooth, bell-shaped velocity profiles of the resulting motions resembled those of humans, verifying the biological relevance of the controller. Notably, the method achieved state-of-the-art control performance while decreasing the motion jerk by 19% to improve motion smoothness.