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In this article we propose a reactive constrained navigation scheme, with embedded obstacles avoidance for an Unmanned Aerial Vehicle (UAV), for enabling navigation in obstacle-dense environments. The proposed navigation architecture is based on Nonlinear Model Predictive Control (NMPC), and utilizes an on-board 2D LiDAR to detect obstacles and translate online the key geometric information of the environment into parametric constraints for the NMPC that constrain the available position-space for the UAV. This article focuses also on the real-world implementation and experimental validation of the proposed reactive navigation scheme, and it is applied in multiple challenging laboratory experiments, where we also conduct comparisons with relevant methods of reactive obstacle avoidance. The solver utilized in the proposed approach is the Optimization Engine (OpEn) and the Proximal Averaged Newton for Optimal Control (PANOC) algorithm, where a penalty method is applied to properly consider obstacles and input constraints during the navigation task. The proposed novel scheme allows for fast solutions, while using limited on-board computational power, that is a required feature for the overall closed loop performance of an UAV and is applied in multiple real-time scenarios. The combination of built-in obstacle avoidance and real-time applicability makes the proposed reactive constrained navigation scheme an elegant framework for UAVs that is able to perform fast nonlinear control, local path-planning and obstacle avoidance, all embedded in the control layer.