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arxiv: 2606.06896 · v1 · pith:QGVTMAHYnew · submitted 2026-06-05 · 🧮 math.OC

Successive Convexification for Trajectory Optimization with Continuous-time Satisfaction of Signal Temporal Logic Specifications

Samet Uzun , Beh\c{c}et A\c{c}{\i}kme\c{s}e This is my paper

Pith reviewed 2026-06-27 21:31 UTC · model grok-4.3

classification 🧮 math.OC
keywords trajectory optimizationsignal temporal logicsuccessive convexificationcontinuous-time specificationsoptimal controlrobustness measuresproximal methods
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The pith

Generalized mean-based robustness lets successive convexification enforce continuous-time signal temporal logic on trajectories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a successive convexification method that incorporates continuous-time signal temporal logic specifications directly into trajectory optimization. It constructs smooth constraints by embedding temporal aggregation into augmented continuous-time dynamics using generalized mean-based robustness, then solves the resulting program with a prox-convex sequential convex programming algorithm that guarantees convergence. This approach is designed to work with time dilation for free final time, multiple-shooting discretization, and finite-dimensional control parameterization. A reader would care because it turns hard-to-handle time-based rules such as always, eventually, and until into forms that standard gradient-based solvers can manage without the locality and gradient-masking problems of ordinary quantitative semantics.

Core claim

The central claim is that generalized mean-based robustness provides a smooth and exact parameterization of discrete-time signal temporal logic that can be lifted to continuous time by embedding the required temporal aggregation into augmented system dynamics; when combined with time dilation, multiple-shooting, and the prox-convex algorithm, the resulting nonconvex optimal-control problem can be solved reliably, yielding trajectories whose continuous-time paths satisfy the original signal temporal logic formulas up to the accuracy set by a user-chosen regularization parameter.

What carries the argument

Generalized mean-based robustness (GMSR) embedded into augmented continuous-time dynamics as the logical building block for differentiable CT-STL constraints.

If this is right

  • The augmentation construction remains largely independent of the particular dynamics transcription or discretization scheme.
  • Accuracy of the resulting CT-STL parameterization is controlled by a single user-selected regularization parameter.
  • The GMSR-based constraints mitigate locality and gradient-masking behavior that appears with standard quantitative semantics.
  • The same framework applies to combined always, eventually, until, and implication specifications on both double-integrator and 6-DoF quadrotor models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The regularization parameter creates an explicit accuracy-versus-smoothness dial that could be adapted online if computation time permits.
  • Because the construction is described as transcription-independent, it may transfer to other optimal-control toolboxes with only local changes to the state vector.
  • If the prox-convex solves remain fast, the method could support receding-horizon replanning under time-varying STL rules.

Load-bearing premise

Embedding temporal aggregation into augmented continuous-time dynamics via GMSR produces an optimization landscape in which the prox-convex method converges to trajectories that satisfy the original continuous-time STL specifications rather than only the smoothed surrogate.

What would settle it

An optimized trajectory that meets all GMSR-augmented constraints yet violates at least one original continuous-time STL formula when evaluated on the full continuous path would falsify the satisfaction guarantee.

Figures

Figures reproduced from arXiv: 2606.06896 by Beh\c{c}et A\c{c}{\i}kme\c{s}e, Samet Uzun.

Figure 1
Figure 1. Figure 1: shows the optimized trajectory for the always obstacle-avoidance example. The vehicle departs from the straight start-to-goal path, bends around the two implication-type forbidden regions, and passes through the admissible corridor before reaching the terminal state. The color-coded speed profile along the trajectory also indicates a smooth acceleration and deceleration pattern, with no abrupt maneuvers re… view at source ↗
Figure 2
Figure 2. Figure 2: Obstacle-clearance histories for the always example. Both clearance signals remain in the safe region throughout the maneuver, confirming continuous-time satisfaction of the two obstacle-avoidance constraints. confirms this behavior quantitatively. The obstacle￾clearance signals remain in the safe region for the entire horizon, so the trajectory never enters either for￾bidden set. The common continuous-tim… view at source ↗
Figure 3
Figure 3. Figure 3: Optimized trajectory for the three-waypoint [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distances from the vehicle to the three waypoint cen [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Optimized trajectory for the charging-station [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Speed profile for the charging-station until example. The speed remains below the pre-charging threshold vsa f e until the charging event occurs, while also remaining below the global speed bound throughout the maneuver. maximize x φunt(·),u(·) − wηηp(tf ) + Γbunt subject to . r(t) = v(t), . v(t) = u(t) − g0e3, . ηp (t) = χθ (u(t)) + χT(u(t)) + χv(v(t)), . y(t) = 1 tf h v 2 sa f e − ∥v(t)∥ 2 2 i2 − , . z(t… view at source ↗
Figure 7
Figure 7. Figure 7: Signed charging-station margin for the until exam￾ple. The margin is negative outside the charging region, zero on its boundary, and positive inside. The first zero crossing marks the witness time for the charging event. 4.2 Free-final-time 6-DoF quadrotor example For the main example, the physical quadrotor state and control are xq := (r, v, ϕ, θ, ψ, pb , qb ,rb ) ∈ R12 , uq := (τϕ, τθ , τψ, T) ∈ R4 , whe… view at source ↗
Figure 8
Figure 8. Figure 8: Optimized 3D trajectory for the free-final-time 6-DoF [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Representative state and input histories for the [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

This paper presents a successive convexification framework for trajectory optimization under continuous-time Signal Temporal Logic (CT-STL) specifications. The framework employs generalized mean-based robustness (GMSR), a smooth and exact parameterization of discrete-time STL, as a logical building block for constructing differentiable CT-STL constraints in optimal control. It is integrated with time-dilation for free-final-time problems, finite-dimensional control parameterization, multiple-shooting discretization of the dynamics, and a convergence-guaranteed sequential convex programming method, prox-convex, to solve the nonconvex program. The main CT-STL realization embeds temporal aggregation into augmented continuous-time dynamics. This augmentation-based construction is largely transcription-independent, can be incorporated into existing optimal-control pipelines with minimal structural changes, and enables smooth CT-STL parameterizations with accuracy controlled by a user-selected regularization parameter. We also discuss a complementary dense-time realization that evaluates CT-STL formulas directly on the integration subnodes used for dynamics discretization, yielding a smooth and exact parameterization on the numerical trajectory representation, up to the accuracy of the integration scheme. The proposed GMSR-based formulations mitigate the locality and gradient-masking behavior of standard quantitative semantics and therefore provide a favorable landscape for gradient-based trajectory optimization. The framework is demonstrated through trajectory-optimization examples for a double-integrator system with continuous-time \always{}, \eventually{}, and \until{} specifications, and a 6-DoF quadrotor flight problem with combined \always{}, \implication{}, and \eventually{}-type specifications. The implementation is available at https://github.com/UW-ACL/TrajOpt_CT-STL.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. This paper presents a successive convexification framework for trajectory optimization subject to continuous-time Signal Temporal Logic (CT-STL) specifications. It employs generalized mean-based robustness (GMSR) as a smooth parameterization of STL, embedding temporal aggregation into augmented continuous-time dynamics for differentiable constraints. The approach integrates time-dilation for free-final-time problems, finite-dimensional control parameterization, multiple-shooting discretization, and the prox-convex sequential convex programming solver. Complementary dense-time and augmentation-based realizations are discussed, with demonstrations on double-integrator systems (always, eventually, until) and a 6-DoF quadrotor (combined always, implication, eventually), and code released on GitHub.

Significance. If the central claims hold, the work provides a largely transcription-independent method to incorporate CT-STL into existing optimal-control pipelines while mitigating locality and gradient-masking issues of standard quantitative semantics. The reproducible implementation and concrete examples on standard benchmark systems strengthen its potential utility for robotics applications requiring temporal logic constraints.

major comments (2)
  1. [Abstract] Abstract: The central claim that the GMSR-augmented dynamics yield an 'exact' parameterization (up to integration accuracy) whose solutions satisfy the original continuous-time STL specifications rests on the regularization parameter controlling accuracy. No analysis is supplied showing that solutions of the regularized problem converge to exact satisfaction of the unsmoothed CT-STL semantics in the limit as the regularization parameter tends to zero, nor that prox-convex SCP on the surrogate avoids local minima that violate the original specification.
  2. [Abstract] Abstract and § on GMSR-based formulations: The assertion that the augmentation-based construction mitigates gradient-masking and provides a favorable landscape for gradient-based optimization is load-bearing for the framework's advantage over standard quantitative semantics, yet the manuscript provides no supporting error bounds, landscape analysis, or comparison of convergence rates to the original (non-surrogate) problem.
minor comments (1)
  1. The GitHub repository link is a positive feature for reproducibility; ensure the released code includes the exact regularization values and integration tolerances used in the reported quadrotor and double-integrator examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address the major comments point by point below, acknowledging the points where the manuscript lacks supporting analysis and outlining planned revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the GMSR-augmented dynamics yield an 'exact' parameterization (up to integration accuracy) whose solutions satisfy the original continuous-time STL specifications rests on the regularization parameter controlling accuracy. No analysis is supplied showing that solutions of the regularized problem converge to exact satisfaction of the unsmoothed CT-STL semantics in the limit as the regularization parameter tends to zero, nor that prox-convex SCP on the surrogate avoids local minima that violate the original specification.

    Authors: The referee correctly notes that the manuscript does not supply a formal analysis of convergence of the regularized solutions to the original CT-STL semantics as the regularization parameter approaches zero. The GMSR is presented as an exact smooth parameterization for discrete-time STL, with the continuous-time embedding relying on this property and the regularization for smoothness. The prox-convex SCP guarantees convergence to a stationary point of the surrogate problem, but not necessarily to a solution satisfying the unsmoothed specification. We will revise the abstract and the relevant sections to qualify the 'exact' claim as holding up to the regularization and integration accuracy, and add a discussion noting the lack of convergence guarantees to the original semantics as an open question. revision: yes

  2. Referee: [Abstract] Abstract and § on GMSR-based formulations: The assertion that the augmentation-based construction mitigates gradient-masking and provides a favorable landscape for gradient-based optimization is load-bearing for the framework's advantage over standard quantitative semantics, yet the manuscript provides no supporting error bounds, landscape analysis, or comparison of convergence rates to the original (non-surrogate) problem.

    Authors: We agree that no quantitative error bounds, landscape analysis, or convergence rate comparisons are included in the manuscript. The claim is motivated by the fact that GMSR replaces the min and max operators of standard quantitative semantics with generalized means, which are differentiable and do not exhibit the same gradient masking in regions where the robustness is large. The augmentation embeds this into the dynamics for a transcription-independent approach. We will revise the abstract and the GMSR section to provide a more detailed qualitative explanation of the expected benefits and acknowledge that a rigorous landscape analysis is beyond the scope of the current work. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces an augmentation-based embedding of GMSR into continuous-time dynamics for CT-STL constraints, combined with standard successive convexification (prox-convex SCP), time-dilation, and multiple-shooting. No load-bearing step reduces by definition or self-citation to its own inputs; GMSR is presented as a building block with explicit regularization parameter, and claims are supported by external quadrotor/double-integrator examples plus public code. Self-citations on SCP (if present) are not invoked as uniqueness theorems or to close the central argument. The framework remains transcription-independent by construction and does not rename known results or smuggle ansatzes.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework relies on a user-selected regularization parameter for smoothing accuracy and standard assumptions from optimal control (convexity of subproblems, differentiability after smoothing). No new invented entities are introduced.

free parameters (1)
  • regularization parameter
    Controls accuracy of the smooth CT-STL parameterization; user-selected and affects the approximation quality.
axioms (1)
  • domain assumption prox-convex sequential convex programming guarantees convergence for the formulated nonconvex program
    Invoked to solve the resulting program after GMSR embedding and discretization.

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