  Symbolic control problems aim to synthesize control policies for dynamical
systems under complex temporal specifications. For such problems, Signal
Temporal Logic (STL) is increasingly used as the formal specification language
due to its rich expressiveness. Moreover, the degree of satisfaction of STL
specifications can be evaluated using ``STL robust semantics'' as a scalar
robustness measure. This capability of STL enables transforming a symbolic
control problem into an optimization problem that optimizes the corresponding
robustness measure. However, since these robustness measures are non-smooth and
non-convex, exact solutions can only be computed using computationally
inefficient mixed-integer programming techniques that do not scale well.
Therefore, recent literature has focused on using smooth approximations of
these robustness measures to apply scalable and computationally efficient
gradient-based methods to find local optima solutions. In this paper, we first
generalize two recently established smooth robustness measures (SRMs) and two
new ones and discuss their strengths and weaknesses. Next, we propose ``STL
error semantics'' to characterize the approximation errors associated with
different SRMs under different parameter configurations. This allows one to
sensibly select an SRM (to optimize) along with its parameter values. We then
propose ``STL gradient semantics'' to derive explicit gradients of SRMs leading
to improve computational efficiency as well as accuracy compared to when using
numerically estimated gradients. Finally, these contributions are highlighted
using extensive simulation results.
