Time Optimization Policy (TOP)

• Time Optimization Policy (TOP) is a family of methods that optimize temporal control as the primary target under domain-specific constraints such as safety, stability, and feasibility.
• Common formulations use a temporal control variable optimized alongside a feasibility condition, applicable in areas from robotic path tracking to test-time adaptation.
• Empirical results show that TOP approaches yield enhanced performance and efficiency, as demonstrated in robotics, reinforcement learning, and dynamic dispatch scenarios.

Searching arXiv for the cited TOP-related papers and closely related work. Time Optimization Policy (TOP) is used in recent literature to denote policies that optimize a temporal degree of freedom under explicit structural constraints. In the cited works, the optimized variable is not uniform across domains: it can be a path-parameterization along a geometric path, the convergence time to an equilibrium under sampling-based Lyapunov stability, inter-clip timestamps for humanoid upper-body motion, the number of self-generated rollouts used for test-time adaptation, or dispatch timing in dynamic team orienteering (Fujii et al., 2023, Wang et al., 2023, Chen et al., 1 Aug 2025, Wang et al., 2 Dec 2025, Wu et al., 16 Jan 2026). This suggests that TOP is best understood as a family of constrained time-optimization policies rather than a single canonical algorithm.

1. Common formulation pattern

Across the cited literature, TOP formulations share a recurrent structure: a temporal control variable is optimized jointly with a feasibility condition that is domain-specific. In robotics path tracking, the objective is to track a geometric path in minimal time subject to kinodynamic constraints and SSM safety. In optimal-time stability, the objective is to minimize discounted cumulative cost subject to mean-cost stability. In humanoid standing manipulation, TOP adjusts timestamps $\Delta t_t$ so as to keep the system within the balance margin while minimizing time. In test-time policy optimization, rollout sampling is halted once posterior confidence exceeds a specified threshold and the retained rollouts are used for an on-policy update. In dynamic team orienteering, the platform decides which requests to accept and how to route workers under hard time budgets (Fujii et al., 2023, Wang et al., 2023, Chen et al., 1 Aug 2025, Wang et al., 2 Dec 2025, Wu et al., 16 Jan 2026).

Domain	Temporal decision	Governing constraint
ISO-safe path tracking	Path traversal time	Kinodynamic constraints and SSM safety
Optimal-time stability	Convergence time in a mean-cost sense	Sampling-based Lyapunov stability
Humanoid standing manipulation	Inter-clip timestamps $\Delta t_t$	Balance, precision, and time efficiency
Test-time policy optimization	Rollout stopping time	Wald thresholds and minimum retained samples
Dynamic team orienteering	Rolling-horizon dispatch timing	Worker time budgets and task time windows

This common pattern does not erase substantive differences among formulations. In some cases TOP is a control policy over physical motion; in others it is a sequential decision rule over compute allocation or online dispatch. The unifying element is that the temporal decision is itself the optimization target, rather than a fixed schedule imposed externally.

2. Reachability-based time-optimal path tracking and ISO safety

A central robotics lineage for TOP is Time-Optimal Path Parameterization (TOPP). In the reachability-analysis formulation of Pham and Pham, a robot with configuration $q\in\mathbb R^n$ follows a geometric path $q(s)$, $s\in[0,s_{\rm end}]$, with time-parameterization $s(t)$ under second-order constraints of the form

$a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$

After discretization into $N$ segments with $x_i=\dot s_i^2$, $u_i=\ddot s_i$, and $\Delta t_t$0, the dynamics become

$\Delta t_t$1

and the algorithm computes reachable and controllable sets by solving small LPs before performing a greedy forward pass. TOPP-RA has two passes, each $\Delta t_t$2 steps with two LPs per step, returns a feasible solution whenever one exists, and is asymptotically optimal as the grid is refined (Pham et al., 2017).

The ISO-safe extension formulates a stricter problem. Given a geometric path $\Delta t_t$3 in configuration space, kinodynamic constraints, dynamic obstacles of known maximum Cartesian speed $\Delta t_t$4, and a protective separation distance $\Delta t_t$5, the objective is to track $\Delta t_t$6 in minimal time subject to kinodynamic constraints and the SSM requirement that, if a collision occurs, the robot must be in a stationary state at the time instant of collision (Fujii et al., 2023). The formulation inherits the TOPP-RA discretization,

$\Delta t_t$7

and adds stoppable sets

$\Delta t_t$8

together with the separation-time quantity

$\Delta t_t$9

At execution time, with current stage-velocity index $q\in\mathbb R^n$0 and $q\in\mathbb R^n$1, the policy selects

$q\in\mathbb R^n$2

and applies the first control from the standard TOPP-RA forward pass on $q\in\mathbb R^n$3 (Fujii et al., 2023).

The theoretical claim is precise: for any robot motion that is strictly faster than the motion recommended by the policy, there exists a human motion that results in a collision with the robot in a non-stationary state. The proof uses two theorems. The first establishes nested stoppable sets, $q\in\mathbb R^n$4 for any $q\in\mathbb R^n$5 and all $q\in\mathbb R^n$6. The second shows that any strictly larger $q\in\mathbb R^n$7 either violates $q\in\mathbb R^n$8 or drives $q\in\mathbb R^n$9, in which case an adversarial obstacle moving at $q(s)$0 can force a collision at stage $q(s)$1 while the robot still has $q(s)$2 (Fujii et al., 2023).

This formulation also clarifies a frequent misconception in safety-critical robotics. In general, it is impossible to avoid all human-robot collisions; the guarantee is instead that if a collision ever occurs, then the robot must have zero joint velocities at that instant. The controller is therefore time-optimal under the stated safety semantics, not a collision-elimination policy. In simulation, it is strictly less conservative than state-of-the-art safe robot control methods. For the 6-DoF robot simulation with a path from RRT, smoothed, $q(s)$3 stages, $q(s)$4, dynamic obstacles at $q(s)$5, and protective distance $q(s)$6, the reported pre-computation improves from $q(s)$7 to $q(s)$8 using CPU multi-core plus GPU LP, while cycle time is $q(s)$9 on CPU and $s\in[0,s_{\rm end}]$0 on GPU; the robot tracks the path, stops exactly at collision, then restarts, and never violates joint limits (Fujii et al., 2023).

3. Optimal-time stability in reinforcement learning

A different TOP formulation appears in model-free reinforcement learning as “optimal-time stability.” The problem is posed on an MDP $s\in[0,s_{\rm end}]$1, where mean-cost stability requires

$s\in[0,s_{\rm end}]$2

and the optimization target is

$s\in[0,s_{\rm end}]$3

The paper’s interpretation is that minimizing the discounted cumulative cost under the stability constraint forces the system to reach equilibrium in the smallest possible cost-weighted time (Wang et al., 2023).

The theoretical device is a sampling-based Lyapunov theorem. It assumes a function $s\in[0,s_{\rm end}]$4 and constants $s\in[0,s_{\rm end}]$5 such that

$s\in[0,s_{\rm end}]$6

$s\in[0,s_{\rm end}]$7

and, in stationarity,

$s\in[0,s_{\rm end}]$8

Under these conditions, the policy satisfies mean-cost stability. The paper further connects the strictness of the condition to a discrete analogue of finite-time stability (Wang et al., 2023).

The practical algorithm is Adaptive Lyapunov-based Actor-Critic (ALAC). It parameterizes a Lyapunov critic $s\in[0,s_{\rm end}]$9, uses a TD-style update toward

$s(t)$0

and defines a constraint-violation measure

$s(t)$1

A single Lagrange multiplier $s(t)$2 enforces $s(t)$3, with adaptive updates $s(t)$4 and $s(t)$5 (Wang et al., 2023).

Empirically, ALAC is evaluated on ten robotic control tasks, including Cartpole, Point-Circle, Swimmer, HalfCheetah, Ant, Humanoid, PyBullet Minitaur, and three free-floating space-robot planning tasks. Against SAC-cost, SPPO, LAC/LAC*, POLYC, LBPO, and TNLF, it attains the lowest cost and near-zero stability violation on all ten tasks. Under persistent action disturbance up to $s(t)$6 control range, its performance degrades far less than baselines, and under desired-goal shifts of $s(t)$7, only ALAC with error in state maintains near-optimal cost (Wang et al., 2023). In this setting, TOP does not mean unconstrained minimum-time motion; it means policy optimization toward the fastest admissible stabilization compatible with the Lyapunov condition.

4. Timestamp optimization for humanoid standing manipulation

In humanoid standing manipulation, TOP is introduced as a three-part framework consisting of a motion prior, a decoupled upper-/lower-body controller, and a Time Optimization Policy. The motion prior is a $s(t)$8-VAE trained on short upper-body motion clips

$s(t)$9

with $a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$0, stacked as $a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$1 and $a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$2. The encoder maps $a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$3 to $a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$4, samples $a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$5, and the decoder reconstructs $a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$6. The loss is

$a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$7

with $a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$8, four 1D-convolution layers with LayerNorm and ReLU, latent dimension $a(s)\,\ddot s + b(s)\,\dot s^2 + c(s)\in C(s).$9, and training for $N$0 epochs with batch $N$1, learning rate $N$2, and cyclical KL schedule (Chen et al., 1 Aug 2025).

Control is then decoupled. The upper body uses a PD controller,

$N$3

while the lower body uses a goal-conditioned policy $N$4 with $N$5 and

$N$6

The lower-body policy outputs $N$7 and is trained with PPO using reward terms that penalize base linear and angular velocities, penalize COM drift through projected gravity, encourage “standing still” leg posture, impose foot contact and slip penalties, and regularize action rate and torque; convergence takes approximately $N$8 hours on RTX4090 (Chen et al., 1 Aug 2025).

TOP acts on timing rather than torque. At each step $N$9, given $x_i=\dot s_i^2$0 and a history embedding $x_i=\dot s_i^2$1, the TOP network $x_i=\dot s_i^2$2 outputs a horizon

$x_i=\dot s_i^2$3

which is smoothed by exponential weighting

$x_i=\dot s_i^2$4

to yield the actual $x_i=\dot s_i^2$5 applied at step $x_i=\dot s_i^2$6. The clip is then replaced by

$x_i=\dot s_i^2$7

with $x_i=\dot s_i^2$8. Training combines supervision and PPO,

$x_i=\dot s_i^2$9

$u_i=\ddot s_i$0

with $u_i=\ddot s_i$1 and $u_i=\ddot s_i$2 (Chen et al., 1 Aug 2025).

At run time, the loop is

$u_i=\ddot s_i$3

The paper states that this closed-loop guarantees whole-body consistency and that TOP actively keeps $u_i=\ddot s_i$4 in a regime where the Zero-Moment Point or Zero-Moment Line remains inside the support polygon; empirically, the ZMP-distance satisfies $u_i=\ddot s_i$5 at all times (Chen et al., 1 Aug 2025). On $u_i=\ddot s_i$6 clips in simulation, the reported comparison is: Exbody, whole-body RL, TimeCost $u_i=\ddot s_i$7, success rate $u_i=\ddot s_i$8, JPE $u_i=\ddot s_i$9, EEPE $\Delta t_t$00, ZMPProj $\Delta t_t$01; Mobile-TV, decoupled, TimeCost $\Delta t_t$02, success rate $\Delta t_t$03, JPE $\Delta t_t$04, EEPE $\Delta t_t$05, ZMPProj $\Delta t_t$06; Ours (TOP), TimeCost $\Delta t_t$07, success rate $\Delta t_t$08, JPE $\Delta t_t$09, EEPE $\Delta t_t$10, ZMPProj $\Delta t_t$11 (Chen et al., 1 Aug 2025). Inference overhead is reported as VAE $\Delta t_t$12, RL policy $\Delta t_t$13, and TOP policy $\Delta t_t$14, for a total of approximately $\Delta t_t$15 at $\Delta t_t$16 on RTX4090 or embedded CPU.

5. Adaptive rollout allocation in test-time policy optimization

OptPO instantiates TOP in a non-physical setting: test-time adaptation of LLMs. The setup assumes a prompt $\Delta t_t$17, a policy $\Delta t_t$18, and an extractor $\Delta t_t$19 mapping each sampled completion $\Delta t_t$20 to one of $\Delta t_t$21 discrete answer tokens $\Delta t_t$22. The goal is twofold: efficiently decide on a consensus pseudo-label by sequentially sampling rollouts $\Delta t_t$23, and use the retained samples for a single on-policy update via PPO, GRPO, or related methods, without requiring ground-truth labels (Wang et al., 2 Dec 2025).

The core mechanism is a Bayesian sequential probability ratio test between the current leader $\Delta t_t$24 and runner-up $\Delta t_t$25. Under hypothesis $\Delta t_t$26, each vote equals $\Delta t_t$27 with probability $\Delta t_t$28 and other classes absorb total probability $\Delta t_t$29. With vote counts $\Delta t_t$30, the likelihood under $\Delta t_t$31 is

$\Delta t_t$32

and the posterior follows by Bayes’ rule. The Bayes-factor is

$\Delta t_t$33

with

$\Delta t_t$34

Given type-I and type-II error budgets $\Delta t_t$35, Wald thresholds are

$\Delta t_t$36

and the equivalent stopping gaps are

$\Delta t_t$37

Sampling stops as soon as $\Delta t_t$38 or $\Delta t_t$39, otherwise continuing until a hard cap $\Delta t_t$40 (Wang et al., 2 Dec 2025).

After stopping at $\Delta t_t$41, the method retains at least $\Delta t_t$42 samples and assigns them the consensus pseudo-label $\Delta t_t$43. A PPO-style update uses

$\Delta t_t$44

where

$\Delta t_t$45

and $\Delta t_t$46 is the $\Delta t_t$47 reward or a smoothed version (Wang et al., 2 Dec 2025).

The efficiency claim is classical and specific. Under simple two-point models, SPRT minimizes the average number of rollouts for fixed type-I and type-II errors, and the expected stopping time is approximated by

$\Delta t_t$48

Empirically, on AIME, AMC, MATH-500, and GPQA with Qwen-Math-1.5B, Qwen-7B, and Llama-3.2-1B backbones, OptPO achieves $\Delta t_t$49–$\Delta t_t$50 token savings versus fixed-budget TTRL baselines at equal or better mean@16 accuracy. The reported examples include GPQA with Qwen-Math-1.5B+PPO, where TTRL-PPO uses $\Delta t_t$51 tokens with mean@16 $\Delta t_t$52 while OptPO-PPO uses $\Delta t_t$53 tokens with mean@16 $\Delta t_t$54, and MATH-500 with Qwen-7B+GRPO, where mean@16 is approximately $\Delta t_t$55 while tokens drop from $\Delta t_t$56 to $\Delta t_t$57 (Wang et al., 2 Dec 2025). Here, TOP refers to optimal rollout allocation, not to trajectory retiming in physical space.

6. Rolling-horizon dispatch, virtual lookahead, and interpretive boundaries

In dynamic team orienteering for spatial crowdsourcing, TOP appears as an event-driven rolling-horizon policy for the Dynamic Team Orienteering Problem in Spatial Crowdsourcing (DTOP-SC). Workers travel along fixed origin-destination trips under hard time budgets, tasks arrive online with release times and time windows, and at each decision epoch the dispatcher solves a deterministic static subproblem over currently available tasks and idle workers. The static snapshot is an HT-TOPTW with binary routing variables $\Delta t_t$58, service indicators $\Delta t_t$59, and service start times $\Delta t_t$60, maximizing

$\Delta t_t$61

subject to route-continuity and time-window constraints (Wu et al., 16 Jan 2026).

The scenario-sampling rolling-horizon framework mitigates myopic bias by augmenting each planning epoch with sampled virtual tasks. At each epoch, $\Delta t_t$62 independent futures are generated: virtual task locations are sampled in the bounding box of current task and worker locations, profits and durations are drawn from empirical ranges, and time-window bounds are drawn from $\Delta t_t$63. For each scenario, the augmented task set $\Delta t_t$64 is solved by ALNS, candidate real tasks are extracted from the resulting routes, and conflict-free assignments are selected from the union of candidates using frequencies $\Delta t_t$65 and the threshold $\Delta t_t$66 with $\Delta t_t$67 (Wu et al., 16 Jan 2026).

The static subproblems are solved with an ALNS à la Ropke–Pisinger using random removal, worst-cost removal, and Shaw removal; greedy insertion, regret-2 insertion, and regret-3 insertion; local search by intra-route 2-opt, inter-route relocate, and inter-route swap; adaptive operator weighting; and simulated annealing acceptance. The reported parameter settings are $\Delta t_t$68, $\Delta t_t$69, $\Delta t_t$70, $\Delta t_t$71, and $\Delta t_t$72 (Wu et al., 16 Jan 2026).

Computationally, the reported results separate benchmark and map-based settings. On the DTOP benchmark with high dynamism, Scen-RH-ALNS obtains mean profit $\Delta t_t$73 versus $\Delta t_t$74 for MPAd and $\Delta t_t$75 for MPAc, with gaps $\Delta t_t$76 versus MPAd and $\Delta t_t$77 versus MPAc, average decision time approximately $\Delta t_t$78, instance-wise best on $\Delta t_t$79 high-dynamism instances and $\Delta t_t$80 overall, and new BKs on $\Delta t_t$81 instances. On map-based DTOP-SC, the Base family reports a gap to the $\Delta t_t$82 MIP incumbent of $\Delta t_t$83 on average with standard deviation $\Delta t_t$84 and time approximately $\Delta t_t$85; scaling from $\Delta t_t$86 to $\Delta t_t$87 workers/tasks, the gap decreases from $\Delta t_t$88 to $\Delta t_t$89 while time grows from approximately $\Delta t_t$90 to approximately $\Delta t_t$91 (Wu et al., 16 Jan 2026).

These formulations delineate the interpretive boundaries of TOP. First, “time-optimal” does not mean “always move or decide as fast as possible”: in ISO-safe tracking, any faster motion than the recommended one admits a human motion leading to a non-stationary collision (Fujii et al., 2023). Second, TOP does not necessarily assume perfect foresight: the ISO-safe controller requires only current minimum distance and obstacle $\Delta t_t$92, not detailed motion prediction; the DTOP-SC policy uses virtual tasks that are never executed and only influence routing; OptPO uses posterior confidence rather than labels from an oracle (Fujii et al., 2023, Wu et al., 16 Jan 2026, Wang et al., 2 Dec 2025). Third, the acronym is domain-specific. Some papers present TOP as a named method, while others contribute key building blocks for what one summary calls the emerging field of Time Optimization Policy (Wang et al., 2023). The common denominator is therefore methodological rather than taxonomic: time is elevated to a controlled resource, and admissibility is encoded through safety, stability, balance, statistical confidence, or routing feasibility constraints.