Authors: Christian Sulana, Christian Hughes, Ian Abraham
Organizations: Christian Sulana is with Saddleback College, Mission Viejo, California, USA · Christian Hughes is with the Department of Mechanical Engineering, Yale University, New Haven, CT 06520, USA · Ian Abraham is with the School of Electrical and Computer Engineering, The University of Sydney, Sydney, Australia
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.
Figures & tables
Fig. 1: Time-Optimal Ergodic Retrieval. A Franka arm executes a full retrieval mission of a green block in an occluded 3D search domain. From its start pose ( t0 ), the arm sweeps the bin along an ergodic trajectory ( t1 ) until the overhead camera detects the target. Once detected, the arm abandons the remaining plan, grasps the object ( t2 ) and delivers it to the base ( t3 ). We propose time-optimal ergodic retrieval (TOER), which includes the expected return cost explicitly in the objective such that searches prioritize likely detection sites that are closer to the base.
Fig. 2: Expected Mission Time Comparison. Single-target search in an open planar field as the sensor’s detection gain is swept at fixed κ=6.6 . When detection is unreliable, the target is usually not found, thus mission time is dominated by return, and TOES + Return suffers additional return cost at every unsuccessful episode. TOER holds a ≥12% advantage over TOES + Return throughout while matching TOES (End @ Base) without imposing a terminal constraint.
Fig. 3: Return Discrepancy Profile. Object search in a planar 12-room lattice. (Left) Search scene with uniform target draws and sensor footprint, occluded by walls. (Right) TOER’s paired mission-time advantage over each baseline vs. return discrepancy κ . Advantage grows logarithmically ( R2=0.95 ) toward the ceiling from Prop. 2 . The gap to TOER (search-only) isolates the return term, which is null at κ=1 and 7.9% at κ=10 .
Fig. 4: Belief Fidelity Impact on Mission. Search in the room setting from Fig. 3 as the true target is displaced by δ from the given belief. (Left) Info-Max’s detection collapses once the belief is wrong ( 0.93→0.22 ) while the ergodic methods degrade slowly, since covering the belief’s support leaves detection mass where the target can still be found. (Right) TOER holds the lowest mission time until the target leaves the belief entirely, where its trade-off between detection speed and terminal position is no longer beneficial.
Fig. 5: Return Bound and Detection Certificate. Validation of the return bound and detection certificate on the non-convex room grid from Fig. 3 . (Left) The two return-cost estimates from one detection site. The obstacle-blind bound cuts through walls while the true geodesic routes around them. (Right, Top) Planning against the loose bound costs 0.1% of realized mission time relative to the geodesic bound. (Right, Bottom) With survival ≤δ imposed, realized survival stays inside the certified region at every δ .
Fig. 6: Computational Scaling of TOER. Log-log wall time for a single solve at fixed iterations ( 95% CI bands on each slope). TOER and TOES scale identically in trajectory knots N , while the Poisson detection term makes TOER grow linearly in domain samples M while TOES is sublinear.