Captivity-Escape Games as a Means for Safety in Online Motion Generation

Abstract

This paper presents a method that addresses the conservatism, computational effort, and limited numerical accuracy of existing frameworks and methods that ensure safety in online model-based motion generation, commonly referred to as fast and safe tracking. Computational limitations restrict online motion planning to low-fidelity models. However, planning with low-fidelity models compromises safety, as the dynamic feasibility of resulting references is not ensured. This potentially leads to unavoidable tracking errors that may cause safety-critical constraint violations. Existing frameworks mitigate this safety risk by augmenting safety-critical constraints in motion planning by a safety margin that prevents constraint violations under worst-case tracking errors. However, the methods employed in these frameworks determine the safety margin based on a heuristically selected performance of the model used for planning, which likely results in overly conservative references. Furthermore, these methods are computationally intensive, and the state-of-the-art method is limited in numerical accuracy. We adopt a different perspective and address these limitations with a method that mitigates conservatism in existing frameworks by adapting the performance of the model used for planning to a given safety margin. Our method achieves numerical accuracy and requires significantly less computation time than existing methods by leveraging a captivity-escape game, which is a novel zero-sum differential game formulated in this paper. We demonstrate our method using a numerical example and compare it to the state of the art.

Figures (3)

• Figure 1: The figure illustrates the task of planning a reference trajectory from $\bm{x}_{ ,\mathrm{l} } (t_{0})$ to $\bm{x}_{ ,\mathrm{l} } (t_{1})$, indicated by the blue arrow. The gray regions indicate unsafe states (e.g., obstacles), the black lines indicate safety-critical constraints, and the green regions indicate the safety margin $\alpha$.
• Figure 2: A valid initial state $\bm{x}_{ , } \left({t_{0}}\right)$ for both games is depicted by the position of PL (respectively, $\tilde{\text{PL}}$). Each game terminates when PL (respectively, $\tilde{\text{PL}}$) exits the green region into the gray region.
• Figure 3: The TEB resulting from the presented method is depicted by the green area. The black circle depicts the boundary of the captivity set $\partial\Lambda$, the green line depicts the IP, and the blue line depicts the closed barrier $\mathcal{K}$. The red dots depict the BIP, the green dots depict $\overline\bm{\nu}(\beta)$, and the gray dots depict where $u_{ ,\mathrm{h} } ^{\diamond}$ switches (see \ref{['eq:sols_up']}). The gray lines illustrate how the numerical fastrack computations evolve, starting in $\partial\Lambda$ and converging to the dotted red line, which depicts the boundary of the TEB determined with the method in fastrack.

Theorems & Definitions (43)

• Definition 1
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• Remark 1
• Definition 9