The vestibular system plays a crucial role in the sense of balance and spatial orientation in mammals. It is a sensory system that detects both rotational and translational motion of the head, via its semicircular canals and otoliths respectively. In this work, we propose a real-time hardware model of an artificial vestibular system, implemented using a custom neuromorphic Very Large Scale Integration (VLSI) multi-neuron chip interfaced to a commercial Inertial Measurement Unit (IMU). The artificial vestibular system is realized with spiking neurons that reproduce the responses of biological hair cells present in the real semicircular canals and otholitic organs. We demonstrate the real-time performance of the hybrid analog-digital system and characterize its response properties, presenting measurements of a successful encoding of angular velocities as well as linear accelerations. As an application, we realized a novel implementation of a recurrent integrator network capable of keeping track of the current angular position. The experimental results provided validate the hardware implementation via comparisons with a detailed computational neuroscience model. In addition to being an ideal tool for developing bio-inspired robotic technologies, this work provides a basis for developing a complete low-power neuromorphic vestibular system which integrates the hardware model of the neural signal processing pathway described with custom bio-mimetic gyroscopic sensors, exploiting neuromorphic principles in both mechanical and electronic aspects.