Abstract
This paper presents a technique, named stlcg, to compute the quantitative semantics of Signal Temporal Logic (STL) formulas using computation graphs. stlcg provides a platform which enables the incorporation of logical specifications into robotics problems that benefit from gradient-based solutions. Specifically, STL is a powerful and expressive formal language that can specify spatial and temporal properties of signals generated by both continuous and hybrid systems. The quantitative semantics of STL provide a robustness metric, i.e., how much a signal satisfies or violates an STL specification. In this work, we devise a systematic methodology for translating STL robustness formulas into computation graphs. With this representation, and by leveraging off-the-shelf automatic differentiation tools, we are able to back-propagate through STL robustness formulas and hence enable a natural and easy-to-use integration with many gradient-based approaches used in robotics. We demonstrate, through examples stemming from various robotics applications, that stlcg is versatile, computationally efficient, and capable of injecting human-domain knowledge into the problem formulation.
K. Leung—This work was supported by the Office of Naval Research (Grant N00014- 17-1-2433) and by the Toyota Research Institute (“TRI”). This article solely reflects the opinions and conclusions of its authors and not of ONR, TRI or any other Toyota entity.
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Notes
- 1.
Equality and the other inequality relations can be derived from the STL grammar in (1), i.e., \(\mu (x) < c \Leftrightarrow -\mu (x) > -c\), and \(\mu (x) = c \Leftrightarrow \lnot (\mu (x) < c) \wedge \lnot (\mu (x) > c)\).
- 2.
This corresponds to padding the input trace with the last value of the robustness trace. A different value could be chosen instead. This applies to Cases 2–4 as well.
- 3.
We can even account for variable signal length by padding the inputs and keeping track of the signal lengths.
- 4.
The \(\epsilon _{ij}\)’s are initialized to zero, which gives negative robustness values and hence results in a non-zero gradient.
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Leung, K., Arechiga, N., Pavone, M. (2021). Back-Propagation Through Signal Temporal Logic Specifications: Infusing Logical Structure into Gradient-Based Methods. In: LaValle, S.M., Lin, M., Ojala, T., Shell, D., Yu, J. (eds) Algorithmic Foundations of Robotics XIV. WAFR 2020. Springer Proceedings in Advanced Robotics, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-030-66723-8_26
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