108 research outputs found
A Forward Reachability Perspective on Robust Control Invariance and Discount Factors in Reachability Analysis
Control invariant sets are crucial for various methods that aim to design
safe control policies for systems whose state constraints must be satisfied
over an indefinite time horizon. In this article, we explore the connections
among reachability, control invariance, and Control Barrier Functions (CBFs) by
examining the forward reachability problem associated with control invariant
sets. We present the notion of an "inevitable Forward Reachable Tube" (FRT) as
a tool for analyzing control invariant sets. Our findings show that the
inevitable FRT of a robust control invariant set with a differentiable boundary
is the set itself. We highlight the role of the differentiability of the
boundary in shaping the FRTs of the sets through numerical examples. We also
formulate a zero-sum differential game between the control and disturbance,
where the inevitable FRT is characterized by the zero-superlevel set of the
value function. By incorporating a discount factor in the cost function of the
game, the barrier constraint of the CBF naturally arises as the constraint that
is imposed on the optimal control policy. As a result, the value function of
our FRT formulation serves as a CBF-like function, which has not been
previously realized in reachability studies. Conversely, any valid CBF is also
a forward reachability value function inside the control invariant set, thereby
revealing the inverse optimality of the CBF. As such, our work establishes a
strong link between reachability, control invariance, and CBFs, filling a gap
that prior formulations based on backward reachability were unable to bridge.Comment: The first two authors contributed equally to this wor
FaSTrack: a Modular Framework for Real-Time Motion Planning and Guaranteed Safe Tracking
Real-time, guaranteed safe trajectory planning is vital for navigation in
unknown environments. However, real-time navigation algorithms typically
sacrifice robustness for computation speed. Alternatively, provably safe
trajectory planning tends to be too computationally intensive for real-time
replanning. We propose FaSTrack, Fast and Safe Tracking, a framework that
achieves both real-time replanning and guaranteed safety. In this framework,
real-time computation is achieved by allowing any trajectory planner to use a
simplified \textit{planning model} of the system. The plan is tracked by the
system, represented by a more realistic, higher-dimensional \textit{tracking
model}. We precompute the tracking error bound (TEB) due to mismatch between
the two models and due to external disturbances. We also obtain the
corresponding tracking controller used to stay within the TEB. The
precomputation does not require prior knowledge of the environment. We
demonstrate FaSTrack using Hamilton-Jacobi reachability for precomputation and
three different real-time trajectory planners with three different
tracking-planning model pairs.Comment: Published in the IEEE Transactions on Automatic Contro
BMQ
BMQ: Boston Medical Quarterly was published from 1950-1966 by the Boston University School of Medicine and the Massachusetts Memorial Hospitals
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