Intelligent robots are designed to effectively navigate dynamic and unpredictable environments laden with moving mechanical elements and objects. Such environment-induced dynamics, including moving obstacles, can readily alter the computational demand (e.g., the creation of new tasks) and the structure of workloads (e.g., precedence constraints among tasks) during runtime, thereby adversely affecting overall system performance. This challenge is amplified when multi-task inference is expected on robots operating under stringent resource and real-time constraints. To address such a challenge, we introduce RED, a systematic real-time scheduling approach designed to support multi-task deep neural network workloads in resource-limited robotic systems. It is designed to adaptively manage the Robotic Environmental Dynamics (RED) while adhering to real-time constraints. At the core of RED lies a deadline-based scheduler that employs an intermediate deadline assignment policy, effectively managing to change workloads and asynchronous inference prompted by complex, unpredictable environments. This scheduling framework also facilitates the flexible deployment of MIMONet (multi-input multi-output neural networks), which are commonly utilized in multi-tasking robotic systems to circumvent memory bottlenecks. Building on this scheduling framework, RED recognizes and leverages a unique characteristic of MIMONet: its weight-shared architecture. To further accommodate and exploit this feature, RED devises a novel and effective workload refinement and reconstruction process. This process ensures the scheduling framework's compatibility with MIMONet and maximizes efficiency.