Control architectures are often implemented in a layered fashion, combining independently designed blocks to achieve complex tasks. Providing guarantees for such hierarchical frameworks requires considering the capabilities and limitations of each layer and their interconnections at design time. To address this holistic design challenge, we introduce the notion of Bezier Reachable Polytopes -- certificates of reachable points in the space of Bezier polynomial reference trajectories. This approach captures the set of trajectories that can be tracked by a low-level controller while satisfying state and input constraints, and leverages the geometric properties of Bezier polynomials to maintain an efficient polytopic representation. As a result, these certificates serve as a constructive tool for layered architectures, enabling long-horizon tasks to be reasoned about in a computationally tractable manner.

翻译：控制架构通常以分层方式实现，将独立设计的模块组合以完成复杂任务。为此类分层框架提供保证需要在设计时考虑各层级的能力、局限及其互连关系。针对这一整体设计挑战，我们提出贝塞尔可达多面体的概念——即贝塞尔多项式参考轨迹空间中可达点的证书。该方法捕获了底层控制器在满足状态和输入约束条件下可跟踪的轨迹集合，并利用贝塞尔多项式的几何特性维持高效的多面体表征。因此，这些证书可作为分层架构的构造性工具，使得长时域任务能够以计算可处理的方式进行推演。