Connections between Reachability and Time Optimality
arxiv(2024)
摘要
This paper presents the concept of an equivalence relation between the set of
optimal control problems. By leveraging this concept, we show that the boundary
of the reachability set can be constructed by the solutions of time optimal
problems. Alongside, a more generalized equivalence theorem is presented
together. The findings facilitate the use of solution structures from a certain
class of optimal control problems to address problems in corresponding
equivalent classes. As a byproduct, we state and prove the construction methods
of the reachability sets of three-dimensional curves with prescribed curvature
bound. The findings are twofold: Firstly, we prove that any boundary point of
the reachability set, with the terminal direction taken into account, can be
accessed via curves of H, CSC, CCC, or their respective subsegments, where H
denotes a helicoidal arc, C a circular arc with maximum curvature, and S a
straight segment. Secondly, we show that any boundary point of the reachability
set, without considering the terminal direction, can be accessed by curves of
CC, CS, or their respective subsegments. These findings extend the developments
presented in literature regarding planar curves, or Dubins car dynamics, into
spatial curves in ℝ^3. For higher dimensions, we confirm that the
problem of identifying the reachability set of curvature bounded paths subsumes
the well-known Markov-Dubins problem. These advancements in understanding the
reachability of curvature bounded paths in ℝ^3 hold significant
practical implications, particularly in the contexts of mission planning
problems and time optimal guidance.
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