Signal temporal logic

Overview

Signal temporal logic (STL) states rules over real-valued signals with time bounds, such as "within two steps, the state must enter a safe set and stay inside it". Each atomic predicate has the form μ(y) − c ≥ 0. Predicates are combined with negation, conjunction, disjunction, and the temporal operators eventually, always and until 2403.15993 §IV-A 2403.15993 eq. 4.

The reason to use STL in a controller is the robustness degree. Instead of a yes or no answer, it returns a signed number: positive when the rule holds, and larger when it holds with more room. Negation flips the sign. Conjunction takes a minimum, disjunction a maximum, eventually a maximum over a window, and always a minimum over a window 2403.15993 tab. 2 2403.15993 §IV-A. Maximising that number gives a controller a graded goal, not just a feasibility check.

Evidence is thin. The one card covers a single robot and reports its results as author-reported. Treat the encoding choices below as options to test, not as settled practice 2403.15993 §VII 2403.15993 §VIII-C.

Variants

Exact mixed-integer encoding

The Boolean structure of the specification is written with binary variables, so the planner is a mixed-integer program. The authors use this as their baseline. It finds the global optimum of the robustness objective in their test, and its solve time grows quickly with the horizon [arxiv:2403.15993 §V-B] 2403.15993 tab. 3. Card: Robust-Locomotion-by-Logic.

Smooth-operator encoding in a nonlinear program

The min and max operators in the robustness are replaced by smooth approximations, each with a positive constant. The result is a nonlinear program that a standard NLP solver can handle. In the authors' test it solves several times faster than the exact encoding, but it reaches a local optimum with lower robustness 2403.15993 §V-B 2403.15993 tab. 3. Card: Robust-Locomotion-by-Logic.

Eventually-within-horizon obligations instead of fixed timing

A recovery event, here the apex of the step, is required to occur somewhere within the horizon, not at a pre-set knot. The paper argues that fixing the timing makes the motion conservative 2403.15993 §V-A. Its recovery condition is stated as a liveness property over the horizon 2403.15993 §III-B. In one ablation case, fixed apex timing fails where the full method recovers 2403.15993 §VIII-F.

When to use which

The timing data come from one robot, one gait, and the authors' solver setup. Do not assume the speed-up carries over to other systems or solvers.

Implementation sketch

The shared core is the same across variants: write the rule, compute its robustness on the predicted trajectory, and add it to the cost. The smooth variant replaces each min with a soft minimum. The paper gives the soft minimum as a log-sum-exp form that under-approximates the minimum for a positive constant 2403.15993 eq. 7 2403.15993 §V-B. The smooth maximum is also named in eq. 7, but the card does not state its exact form, so check the source before copying it.

# Soft minimum of the robustness terms  [arxiv:2403.15993 §V-B] [eq. 7]
def soft_min(rhos, k1):
    # Under-approximates min(rhos) for k1 > 0  [arxiv:2403.15993 §V-B]
    return -log(sum(exp(-k1 * r) for r in rhos)) / k1

# Robustness of the stability clause, as a cost term  [arxiv:2403.15993 §VI-A]
def stability_robustness(traj, horizon_knots):
    # Eventually over the last step is a max over the window  [arxiv:2403.15993 §V-A]
    inner = [soft_min(predicate_margins(y), k1)
             for y in traj[horizon_knots]]
    return soft_max(inner, k2)

# Objective: effort minus robustness  [arxiv:2403.15993 §VI-A] [eq. 8]
# The weight w on effort is small, favouring the specification  [§VII]
def cost(traj, controls, w):
    # Squared swing-foot velocities over all knots  [arxiv:2403.15993 §VI-A]
    effort = sum(u ** 2 for u in controls)
    return w * effort - stability_robustness(traj, horizon_knots)

The constants k1 and k2 are not given in the paper. Choose them and report the values you use 2403.15993 §V-B.

Contradictions

Within the one paper, two reported figures do not match the rest of the text.

• Solve-time ratio. The text says the exact encoding is 5 to 10 times slower 2403.15993 §VIII-C. The mean solve times in the table give about 7.6 times at N = 2, about 17 times at N = 3, and about 19 times at N = 4 2403.15993 tab. 3. Use the table.
• Impulse tolerance. The conclusion claims an 81 % increase in maximum impulse tolerance during leg crossing 2403.15993 §XI. The body shows the comparison only in a figure without tabulated values 2403.15993 fig. 15, so the 81 % figure cannot be checked from the text.

Papers

• Robust-Locomotion-by-Logic: STL robustness as the MPC objective for Cassie push recovery, with exact and smooth encodings compared.