Arithmetic-Geometric Mean Robustness for Control from Signal Temporal Logic Specifications

Abstract

We present a new average-based robustness score for Signal Temporal Logic (STL) and a framework for optimal control of a dynamical system under STL constraints. By averaging the scores of different specifications or subformulae at different time points, our new definition highlights the frequency of satisfaction, as well as how robustly each specification is satisfied at each time point. We show that this definition provides a better score for how well a specification is satisfied. Its usefulness in monitoring and control synthesis problems is illustrated through case studies.

Trajectory Optimization for High-Dimensional Nonlinear Systems Under STL Specifications

“…The scenario shown in Figure 3 represents something of a benchmark problem in the STL planning literature [2], [6], [7]. A robot must visit one of two intermediate targets (blue) and avoid collisions with an obstacle (red) before reaching a goal region (green).…”

Average-based Robustness for Continuous-Time Signal Temporal Logic

“…This example illustrates the need for a continuous-time score, as discretizing the system is not always preferable, especially when an appropriate discretization frequency is not known. [19] and propose a new averagebased robustness η for bounded continuous-time signals that captures more information about the signal relative to the traditional score. Our robustness definition returns a normalized score η ∈ [−1, 1] with η ∈ (0, 1] and η ∈ [−1, 0) corresponding to satisfaction and violation of the specification, respectively; and η = 0 when satisfaction is inconclusive.…”

Coupled Multi-Robot Systems Under Linear Temporal Logic and Signal Temporal Logic Tasks

“…STL entails space robustness [20], a form of the robust semantics [21], stating how robustly a signal satisfies a temporal logic formula. Control synthesis under STL tasks has been considered for single-agent systems [22]- [30] and for multi-agent systems [31], [32]. A common trade-off is to find a restricted, yet expressive STL fragment that allows for computationally-efficient control as in [25], [31].…”

“…The LTL tasks are φ 1 := GF (α 1,1 ∧ α 1,2 ) and φ 2 := GF (α 2,1 ∧ α 2,2 ) or, in words, robot 1 (robot 2) should periodically visit α 1,1 and α 1,2 (α 2,1 and α 2,2 ). For robot 3, define µ 3,1 := ( z 3 − 0 −≤ 0.1), µ 3,2 := ( z 3 − 1.5 −1.≤ 0.1), µ 3,3 := ( z 3 − z 4 ≤ 0.7), φ 3 := G[21,30] µ 3,1 , and φ 3 := F [57,58] (µ 3,2 ∧ µ 3,3 ). Robot 3 is then subject to φ 3 := φ 3 ∧ φ 3 or, in words, always between 21-30 sec be in region µ 3,1 and eventually between 57-58 sec be in region µ 3,2 while being at least 0.7 m close to robot 4.…”