Time-Optimal Ergodic Retrieval

The gist Retrieval tasks are a form of search task where a robot must explore, find an object, and return with it as quickly as possible<ref:2610.21823#pg6> This work proposes a new formulation of the time-optimal ergodic search problem that models detection as a Poisson process and minimizes total mission time by including the return phase in the objective

How it works

The proposed Time-Optimal Ergodic Retrieval (TOER) models detection along a trajectory as an inhomogeneous Poisson process whose rate is set by the sensor’s response to a given target distribution such that the probability that the robot is still searching decays with accumulated detection pressure<ref:2610.21823#pg6> The objective sums the expected search time with an expected return time charged from the node at which detection occurs<ref:2610.21823#pg6> This formulation retains ergodicity as a constraint to preserve the coverage guarantees of time-optimal ergodic search and eliminate the degenerate solution of immediately abandoning the search<ref:2610.21823#pg6>

The objective function is constructed by summing the expected search time with an expected return time charged from the node at which detection occurs<ref:2610.21823#pg6> Segment i contributes its time τi only if the robot is still searching when it begins (Pi−1)<ref:2610.21823#pg6> If the object is first found in segment i (probability πi), the robot returns from xi at cost R(xi)<ref:2610.21823#pg6> If it is never found (probability PN), the robot returns from its final node xN with cost R(xN)<ref:2610.21823#pg6> The total mission time is the sum J(¯x) = X N i=1 τi Pi−1 + X N i=1 πi R(xi) + PN R(xN), where the first term is the expected search time and the remaining two are the expected return time<ref:2610.21823#pg6>

Key Concepts and Guarantees

The formulation minimizes the expected duration of a search-and-return mission subject to an ergodic constraint<ref:2610.21823#pg6> The ergodic constraint demands that the trajectory cover the target I to tolerance γ in the metric of (3)<ref:2610.21823#pg6> This approach avoids the issue where current time-optimal formulations define a terminal boundary constraint to force the trajectory to end at a base, which fixes the final pose even when detection occurs at an earlier segment

The paper derives guarantees that any admissible lower bound on the return yields a certifiably ε-optimal plan, and that ergodic coverage guarantees hold in this formulation<ref:2610.21823#pg6> The value of return-awareness scales predictably with the difficulty of the return toward a computable ceiling fixed by domain geometry, coverage tolerance, and the target distribution<ref:2610.21823#pg6>

Validation and Results

Experimental results show that time-optimal ergodic retrieval (TOER) achieves up to 19% shorter mission durations relative to ergodic baselines and that this advantage grows as the return phase becomes more costly<ref:2610.21823#pg6> Simulation and hardware implementations validate TOER’s compatibility with environments of non-convex geometry as well as both 2D and 3D search domains<ref:2610.21823#pg6>

The advantage grows logarithmically (R2 = 0.95) toward the ceiling from Prop. 2 TOER’s paired mission-time advantage over each baseline vs. return discrepancy κ The margin over TOES (End @ Base) is the smallest because, at unreliable detection, ending at base is close to return-optimal

Computational Scaling and Robustness

The computational cost of TOER grows linearly in domain samples M while TOES is sublinear Solving TOER’s search program without the return term costs 125.6s against TOER’s 126.9s, which verifies that the computational cost is primarily due to the Poisson objective

TOER is robust to prior misspecification, as it is the fastest method at every displacement up to δtarget = 1 and completes the mission in 12.2s at δtarget = 0.7 against TOES’ 13s and Info-Max’s 16.5s Beyond this range, the target leaves the belief entirely and no planner can allocate search time from the prior

The obstacle-blind estimate is at most the geodesic return for 99.3% of sampled states with a mean gap of 0.57s, which confirms the ordering stated in Prop. 1 The remaining 0.7% of samples violate the ordering by at most 0.014 s, which is below the 0.085 s corresponding to one cell diagonal of the geodesic grid

The paper concludes that TOER reduces mission time relative to baselines and yields mission time reductions up to a fixed ceiling as return becomes more costly Future work will extend the method to single-shot planning for multiple retrievals under a given capacity constraint

The gist Retrieval tasks are a form of search task where a robot must explore, find an object, and return with it as quickly as possible This work proposes a new formulation of the time-optimal ergodic search problem that models detection as a Poisson process and minimizes total mission time by including the return phase in the objective

How it works

The proposed Time-Optimal Ergodic Retrieval (TOER) models detection along a trajectory as an inhomogeneous Poisson process whose rate is set by the sensor’s response to a given target distribution such that the probability that the robot is still searching decays with accumulated detection pressure<ref:2610.

1. Bold header: Formulation of Time-Optimal Ergodic Retrieval (TOER) for retrieval missions. This allows AI systems to minimize the expected duration of a search-and-return mission by charging the return from an unknown detection point, which is superior to current planners that minimize the duration of a full deterministic search.

2. Bold header: Integration of Poisson process modeling for detection uncertainty. The system can model detection as a Poisson process and use the resulting likelihoods, such as the survival probability is e−Λi,j, to drive search behavior based on accumulated detection pressure.

3. Bold header: Incorporation of return cost into the objective function. The AI planner will minimize the expected total mission time by summing the expected search time with an expected return time charged from the node at which detection occurs.

4. Bold header: Preservation of ergodic coverage guarantees under retrieval costs. By including the return term in the objective, the planner maintains that ergodic coverage guarantees carry over unchanged, ensuring that the optimized horizon still adheres to a target distribution within tolerance.

5. Bold header: Certification of ε-optimality via admissible return bounds. The system can leverage Proposition 1 (Admissible Return Bound) to generate a plan where the resulting trajectory is ε-optimal, Jκ(¯x†) ≤ J∗κ + ε, providing a computable certificate of performance against any admissible return cost estimate.

6. Bold header: Robustness to prior misspecification via survival constraints. The planner can enforce detection confidence by appending the survival constraint P(tf) ≤ δ of Prop. 3, allowing users to specify a detection confidence and determine the resultant cost in mission time before deployment.

Retrieval tasks are a form of search task where a robot must explore, find an object, and return with it as quickly as possible. Recent work in ergodic control has demonstrated time-optimal path planning for search tasks where visitation is proportionally matched, within some tolerance, to a given target distribution. However, these planners minimize the duration of a full deterministic search, while retrieval missions end their search at an unknown point in the trajectory and must return to a given base. In this work, we propose a new formulation of the time-optimal ergodic search problem that models detection as a Poisson process and minimizes total mission time by including the return phase in the objective. Experimental results show that time-optimal ergodic retrieval (TOER) achieves up to 19% shorter mission durations relative to ergodic baselines and that this advantage grows as the return phase becomes more costly. Simulation and hardware implementations validate TOER's compatibility with environments of non-convex geometry as well as both 2D and 3D search domains.