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Vector Lyapunov functions are a multi-dimensional extension of the more familiar (scalar) Lyapunov functions, commonly used to prove stability properties in systems described by non-linear ordinary differential equations (ODEs). This paper explores an analogous vector extension for so-called barrier certificates used in safety verification. As with vector Lyapunov functions, the approach hinges on constructing appropriate comparison systems, i.e., related differential equation systems from which properties of the original system may be inferred. The paper presents an accessible development of the approach, demonstrates that most previous notions of barrier certificate are special cases of comparison systems, and discusses the potential applications of vector barrier certificates in safety verification and invariant synthesis.
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Guaranteed Safe Path and Trajectory Tracking via Reachability Analysis Using Differential Inequalities
In many automated motion planning systems, vehicles are tasked with tracking a reference path or trajectory that is safe by design. However, due to various uncertainties, real vehicles may deviate from such references, potentially leading to collisions. This paper presents rigorous reachable set bounding methods for rapidly enclosing the set of possible deviations under uncertainty, which is critical information for online safety verification. The proposed approach applies recent advances in the theory of differential inequalities that exploit redundant model equations to achieve sharp bounds using only simple interval calculations. These methods have been shown to produce very sharp bounds at low cost for nonlinear systems in other application domains, but they rely on problem-specific insights to identify appropriate redundant equations, which makes them difficult to generalize and automate. Here, we demonstrate the application of these methods to tracking problems for the first time using three representative case studies. We find that defining redundant equations in terms of Lyapunov-like functions is particularly effective. The results show that this technique can produce effective bounds with computational times that are orders of magnitude less than the planned time horizon, making this a promising approach for online safety verification. This performance, however, comes at the cost of low generalizability, specifically due to the need for problem-specific insights and advantageous problem structure, such as the existence of appropriate Lyapunov-like functions.
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- Award ID(s):
- 1949748
- PAR ID:
- 10431461
- Date Published:
- Journal Name:
- Journal of intelligent robotic systems
- ISSN:
- 1573-0409
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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