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Systems-on-Chips (SoCs) that power autonomous vehicles (AVs) must meet stringent performance and safety requirements prior to deployment. With increasing complexity in AV applications, the system needs to meet these real-time demands of multiple safety-critical applications simultaneously. A typical AV-SoC is a heterogeneous multiprocessor consisting of accelerators supported by general-purpose cores. Such heterogeneity, while needed for power-performance efficiency, complicates the art of task scheduling. In this paper, we demonstrate that hardware heterogeneity impacts the scheduler's effectiveness and that optimizing for only the real-time aspect of applications is not sufficient in AVs. Therefore, a more holistic approach is required -- one that considers global Quality-of-Mission (QoM) metrics, as defined in the paper. We then propose HetSched, a multi-step scheduler that leverages dynamic runtime information about the underlying heterogeneous hardware platform, along with the applications' real-time constraints and the task traffic in the system to optimize overall mission performance. HetSched proposes two scheduling policies: MSstat and MSdyn and scheduling optimizations like task pruning, hybrid heterogeneous ranking and rank update. HetSched improves overall mission performance on average by 4.6x, 2.6x and 2.6x when compared against CPATH, ADS and 2lvl-EDF (state-of-the-art real-time schedulers built for heterogeneous systems), respectively, and achieves an average of 53.3% higher hardware utilization, while meeting 100% critical deadlines for real-world applications of autonomous vehicles. Furthermore, when used as part of an SoC design space exploration loop, in comparison to prior schedulers, HetSched reduces the number of processing elements required by an SoC to safely complete AV's missions by 35% on average while achieving 2.7x lower energy-mission time product.