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SparseMax-dSTL: Sparse Projection Approximation of STL Robustness

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Sai Sandeep Damera et al. proposed the SparseMax-dSTL method, which replaces the commonly used log-sum-exp smooth approximation with a lower approximation based on the sparsemax projection to compute signal temporal logic (STL) robustness. The authors prove that this approximation is always lower than the exact robustness and is therefore sound, and that its error upper bound is independent of the number of samples and can be determined before optimization. In simulations, in tasks involving a planar unicycle and a 7-DOF robotic arm (with a human nearby), the log-sum-exp approximation still gave positive values on trajectories that violated the specification; for the robotic arm task, when the time horizon increased from 10 seconds to 80 seconds, the gradient weight of this approximation on violating samples dropped from 0.22 to 0.04, whereas the sparsemax version remained at 1. In constrained trajectory optimization for the unicycle, the log-sum-exp robustness constraint returned trajectories that violated the specification, whereas the sparsemax constraint returned trajectories that satisfied the specification with margin.