A smoother keeps violating samples in the trajectory update

When a robot briefly breaks a temporal safety rule and then waits, can a trajectory optimizer still detect the violation and keep it influential in its updates?

The answer matters because the optimizer relies on a smoothed score both to judge constraints and to distribute the gradient signal across time samples.

SparseMax-∂STL replaces STL’s minimum and maximum operators with sparse lower estimates, keeping gradients focused on violating samples as horizons grow.

The authors tested whether a smooth trajectory constraint could reject sampled violations without losing their influence on updates as a robot waits longer.

SparseMax-∂STL replaces STL extrema with lower estimates, so a nonnegative smoothed value certifies nonnegative exact robustness on the sampled trajectory.

In four simulated manipulator trajectories, plain log-sum-exp gave positive median robustness despite negative exact robustness. As the horizon grew from 10 to 80 seconds, sparsemax retained all measured gradient weight on violating samples, while sound log-sum-exp’s share fell.

A unicycle optimization also returned positive exact robustness with sparsemax and negative robustness with plain log-sum-exp.

The guarantee concerns sampled STL semantics in exact arithmetic, so it says nothing about behavior between samples.

The optimization evidence comes from simulations and first-order solvers; sparsemax can reject trajectories when its realized error exceeds their exact margin, and its zero-weight threshold can hide samples from the gradient.