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Guaranteed Safe Path and Trajectory Tracking via Reachability Analysis Using Differential Inequalities

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Abstract

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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Data Availability

The datasets generated for this article are obtainable from the corresponding author upon reasonable request.

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Funding

This material is based upon work supported by the National Science Foundation under grant no. 1949748.

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Authors and Affiliations

  1. School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, North Ave NW, Atlanta, GA, 30332, USA

    Xuejiao Yang, Bowen Mu, Dillard Robertson & Joseph Scott

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  1. Xuejiao Yang
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  2. Bowen Mu
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  3. Dillard Robertson
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  4. Joseph Scott
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Contributions

J. Scott and X. Yang contributed to the study conception and design, algorithm development, and theoretical analyses. X. Yang led the software development and simulations with assistance from B. Mu. Comparisons to CORA were run by D. Robertson. The first draft was written by X. Yang and edited by J. Scott. All authors read and approved the final manuscript.

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Correspondence to Joseph Scott.

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Yang, X., Mu, B., Robertson, D. et al. Guaranteed Safe Path and Trajectory Tracking via Reachability Analysis Using Differential Inequalities. J Intell Robot Syst 108, 78 (2023). https://doi.org/10.1007/s10846-023-01928-w

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