Laundry-hanging robots

A 2D cloth-drape model (discrete elastic chain with penalty contact, solved by L-BFGS) scores how far a garment hangs from its worn shape, and CMA-ES designs the hanger. Over seven sizes from a child’s tee to XXL, the best single fixed hanger averages 2.1 cm RMS shape error (5.4 cm worst). One adjustable-width hanger set per garment averages 0.18 cm (0.22 worst), 12× better, and a rack of four widths keeps every size within 1 cm. Wire and plastic hangers fold the cloth at the tip (about 65°/cm against 14° worn); a 2–3 cm round end removes it, but a loop scored on shape alone converges on wire-thin ends. CMA-ES beats random search (0.19 vs 0.44 cm at 150 drapes). Stylised: 2D, no friction, assumed fabric constants.

Adds a stretch-set model for hanging (permanent shoulder lengthening from hanger-fit’s drape) and a crease-set model for folding to five fabric archetypes, prices both in cm of departure from the pressed shape, and learns a rule on 600 simulated garments. Woven shirts and linen are better hung, t-shirts, fleece and heavy knits better folded. Shoulder width, mass and stretch when lifted choose the better action for 89–91% of garments in 5-fold cross-validation (always fold: 59%; adding the fabric class: 94%); width alone adds nothing. The split is driven by the assumed crease model: a 6 h stack instead of 48 h moves the share better hung from 41% to 8%, while a hundredfold change in the stretch penalty moves it from 48% to 37%. Stylised: every fabric constant is assumed, the population is synthetic, no motion.

A swept-section geometry model passes a hanger through a garment’s neck or hem loop: the loop must reach round the hull of the hanger’s cross-section, and a planner finds the path of least stretch. A rigid hanger needs 43–61 cm of loop (a whole arm), which stretches a tee’s neck by up to 33% against an assumed 35% limit, and a buttoned woven collar cannot take it, so every shirt goes in at the hem and has its hook threaded out blind (about 10 s against 6 s). A telescoping hanger collapsed to a 10 cm half width needs 31 cm and passes five of seven shirt sizes; folding the arms down 60° needs 37–40 cm. The 2–3 cm round end that hanger-fit needs adds only about 3 cm of loop, so insertion changes the width design, not the end. Stylised: planar hanger, perimeter bound rather than cloth, assumed stretch limits and robot constants.

A discrete-event model of the whole cell (pick, sense, hanger feeder, insertion, rail and fold table) fed with hanger-insert’s insertion times and hanger-fold’s hang-or-fold shares. With assumed pick and fold times, one arm clears a 25-item dryer load in 8.9 min (169 items/h, about 5 dryers on a 45 min cycle); a second arm adds 22% and then the fold table is the bottleneck. The five hanger designs differ by at most 3% in time per load. Doubling the fold time adds 5.9 min and doubling the pick time 3.4, against under 1 min for insertion, though a threefold slower insertion costs a one-arm cell 22%. A poor picker (0.60 success) costs 13% throughput but leaves 7 of every 100 items for a person. Stylised: every station time except insertion is an assumed placeholder, no layout or collisions.

Exports hanger-fit’s adjustable shape and four-width rack as parametric STL and OpenSCAD, with a cantilever strength, deflection and print-bed check. Printed flat in PLA under assumed constants (1.5 kg garment, 2× motion factor), the round-ended arm (end radius 2.86 cm) has a safety factor of 40–70 at 12 mm depth and bends under 0.04 mm, so strength never limits it; a wire-like arm (end radius 1.5 mm) has a safety factor of 0.13 and fails. No hanger of useful width fits a 220 mm bed whole, but one arm with its hub does up to 24.8 cm. Stylised: assumed material constants, flat extrusion, OpenSCAD files not rendered, no two-piece joint.

A stylised 2D pile of overlapping disc garments, 40% with the opening hidden by the garment itself, emptied by three scripted policies with assumed times. Aiming at a visible neck or waistband takes 13.4 s per garment against 22.7 s for pinching the top and regrasping; pushing garments aside first is slower (17.7 s) because neighbours rarely hide every opening, the garments themselves do. In hanger-cell the policies give 151, 110 and 127 items an hour for one arm, against 169 for its placeholder pick. Stylised: assumed times, disc cloth, not a measurement of a robot.

On paired piles in the stylised hanger-pick model, aiming at a visible opening beats a blind pinch only if finding the opening costs under 9.2 s per pick (95% interval 8.9 to 9.6, assumed times). The threshold grows with the time of a regrasp (2.9 s to 18.7 s for 2 to 12 s), shrinks as regrasps succeed more often (14.7 s to 3.9 s) and as more openings are hidden (15.3 s to 0), and does not depend on pile size. Sweeping garments aside first beats aiming only when a sweep costs under half a second. Stylised: every time is an assumed placeholder.

In the stylised hanger-pick model a wrong aim costs one wasted grasp and then the same regrasp hunt as a blind pinch. The look-time budget of the detector falls almost linearly with its error: 9.2 s per pick when perfect, 6.3 s at 30% wrong, 4.5 s at 50%, about 0 when always wrong (300 paired piles, assumed times). At a 1 s look aiming still wins until about 88% of aims are wrong; at 5 s the limit is 43%, at 9 s 3%, and an 11 s look loses even when perfect. Stylised: every time and probability is an assumed placeholder; error is independent per pick.

A 3D cloth sheet (stretch, shear and bending in both directions, penalty contact with a swept arm) reproduces the 2D hanger-fit drape when the arm is as deep as the garment: shape error within 0.05 cm for the round-ended and wooden hangers and within 0.6 cm for thin wire (which is resolution-sensitive). The width finding holds: the best arm is 1 cm wider than the half shoulder for child, M and XXL tees, in 2D and 3D. What 2D cannot show is depth: on a 3 cm half-depth arm the wooden hanger's whole-sheet error rises from 0.4 to 2.0 cm, and on a rod to 3.8 cm, as the front and back panels fall 5–8 cm below the arm. A CMA-ES design loop rerun in 3D (M tee, 64 drapes, two seeds) shows arm depth sets the whole-sheet error floor (about 2.1, 1.5 and 0.7 cm at 1.5, 3 and 6 cm half-depth) while shape matters less: hangers designed on the 2D centre line are within seed noise of 3D-designed ones. Stylised: assumed fabric constants, one 20 cm strip, coarse grid (0.5 cm by 2.2 cm), no collar, sleeve seam or friction.

hanger-3dseeds left three of eight 3D-designed optima on its 6 cm end-radius cap. Here the cap is lifted to 10 and 15 cm (M tee, 3 cm half-depth arm, 64 drapes per loop, 8 seeds). Whole-sheet error: 1.08 ± 0.29 cm at 10 cm and 1.21 ± 0.32 at 15 cm, against 1.12 ± 0.25 at 6 cm (differences −0.04 cm, p = 0.77, and +0.08 cm, p = 0.57). Chosen radius grows with the cap (mean 8.2 and 10.6 cm) while crease and sag fall slowly (sag 2.9, 2.2, 1.7 cm), so blunter ends only trade a flat error for smoother drape. Stylised: assumed fabric constants, one 20 cm strip, no friction, one garment and arm depth, a small search budget; a 10 to 15 cm tip is not a usable hanger and no contact-width constraint is applied.

hanger-3d found that a 3D design loop beat hangers designed on the 2D centre line by 0.2–0.4 cm at 3 cm half-depth on two seeds, within seed noise. Here are eight seeds per route (M tee, 3 cm half-depth arm, 64 drapes per 3D loop, 150 per 2D loop). Whole-sheet error: 2D-designed 1.80 ± 0.13 cm (mean ± sd), 3D-designed with hanger-fit's bounds 1.54 ± 0.28 (difference −0.26, bootstrap 95% CI −0.44 to −0.03, exact permutation p = 0.033), 3D-designed with bow up to 5 cm and end radius up to 6 cm 1.12 ± 0.25 (−0.68 against 2D, CI −0.85 to −0.49, p < 0.001; −0.42 against the original bounds, p = 0.006). The 3D optimum sits on the radius limit in three of eight wide-bound runs, so the bound still binds. The wide-bound designs also drop the crease from about 25 to 11 deg/cm and the edge sag from 4.6 to 2.9 cm but have a worse centre-line error (0.77 against 0.40 cm), so the 2D score and the whole-sheet score disagree. Raising the radius limit to 9 and 12 cm (8 seeds each) leaves no seed on the limit at 12 cm and changes the error by only −0.10 to −0.11 cm (p > 0.4); the optimum radius ranges 4.8–10.4 cm, so the error is flat in it. Stylised: assumed fabric constants, one 20 cm strip, coarse grid, no friction; one garment and one arm depth; the budget (64 drapes) is small, so a longer search may change the gap (untested).

Two arms on one hanger-pick pile in the hanger-cell model; an arm whose choice would clash with the other arm's current pick (grasps closer than a clearance) picks the best garment that does not, and waits only if none is left. At 0.30 m blind pinching recovers 9.5 of the 28 garments an hour that blocking cost it (151 to 161), sweeping 5 (186 to 191), aiming about 1.6 of 10 (191 to 192, inside its interval). At 0.45 m nearly every pair clashes and it recovers nothing; with avoidance a third arm adds about 7.5 an hour at 0.30 m. Stylised: the clearance, the spacing time and all station times are assumed, not measured.

hanger-avoid left aiming's gain from avoiding a clash unproven. Pairing wait and avoid on the same 2000 seeded loads (two arms, load 25) gives, at 0.30 m clearance, +1.25 garments an hour for aiming (95% bootstrap interval 0.60 to 1.92), +4.4 for sweeping and +10.0 for blind pinching; at 0.45 m aiming gains +0.15 (−0.05 to 0.36). Pairing halves the standard error against unpaired intervals, yet avoiding wins on only about half of single loads for aiming (load-to-load SD about 15 an hour). Stylised: clearance, station times and the pile are hanger-avoid's placeholders; only the statistics are tightened.

Replacing the assumed 8 s, 0.85 pick of the hanger-cell discrete-event model with the hanger-pick pile model cuts a one-arm cell from 169 to 147 garments hung or folded per hour (5.1 to 4.4 dryers) with look-then-aim picking, and to 104 with blind pinching. With two arms the fold table is the limit and the policy matters little (206 assumed, 201 aiming, 179 pinching); quartering all pick times still leaves about 6.2 dryers. Aiming leaves 2.8 garments per 100 for a person, pinching 7.2. Stylised: every station time is an assumed placeholder.

Taking the fold time from Avigal et al. (2022), SpeedFolding (30–40 folds per hour, about 90–120 s each) instead of hanger-cell's assumed 25 s, the one-arm cell with the learned hang-or-fold rule drops from 169 to 58–44 garments an hour (5.1 to 1.8–1.3 dryers) and is fold-bound (table 98% busy, arm 24–31%). Always hanging stays at 146 an hour, 3.3 times the learned rule at 120 s, and beats always folding for any fold time above about 24 s. More fold tables recover throughput only slowly (115 an hour needs 3 at 120 s). Sourced: the fold rate. Assumed and unchanged: every other station time, the load mix and the claim that the fold runs on a table separate from the arm; hanging quality is not scored, so ‘always hang’ is a speed bound, not a recommendation.

The hanger-cell model assumed a 25 s fold station and found the arm was the limit (5.1 dryers). Replacing 25 s with the cycle times in Table I of the SpeedFolding paper (Avigal et al. 2022: 68 s for a towel, 88 s fling-to-fold, 117 s t-shirt, 167 s max-value-map baseline, with 80 to 93% fold success) makes the fold table the bottleneck at 98% busy: one arm and one table serve 1.3 dryers at the t-shirt time (0.8 to 2.1 across the rows). With two arms, five dryers need 3 to 10 fold tables depending on the row (5 at 117 s). Prior art at 3 to 6 folds per hour would need more than 40. Hanging every garment instead of folding doubles output at the same tables, but hang-or-fold is a quality choice (hanger-fold), not a throughput one. Fold times and success rates are sourced from one paper measured on one robot and 15 trials per row; the other station times (pick, sense, rail, feeder) are still assumed placeholders from hanger-cell.

hanger-sweep could not find measured fabric constants and swept them. This run uses a published KES-FB table (Ramaiah et al., fibre2fashion PDF, Table 3) for four fabrics: cotton single jersey, elastane single jersey, cotton shirting and heavy cotton suiting (bending rigidity 0.0066–0.125 gf·cm²/cm, weight 12.5–24.6 mg/cm²). On an M tee shape the best arm half width of the stock hanger is the half shoulder + 1 cm in all 12 cases (4 fabrics × stretch stiffness 40, 150, 800 N/m), best error 0.18–0.83 cm RMS; the suiting at 40 N/m nearly ties with +2 cm (0.83 vs 0.87). Bending rigidity and weight are sourced; stretch stiffness is swept because the table lacks it; the garment geometry and the tee mass scale are hanger-fit's assumptions for every fabric; the source fabrics are unfinished.

Linking the hanger-pick pile's garment kinds to the hanger-cell archetypes moves a one-arm laundry cell from 147 to 141 garments hung or folded per hour (look-then-aim), but the effect on two arms depends on an assumed map from pile kinds to archetypes: 141 to 250 per hour across five maps. The correlation the link adds is small beside that: raising a knit's chance of hiding its opening from 0.2 to 1.0 costs 144 to 132 per hour on one arm and sets 0.23 knits per load aside. Stylised: station times, pile probabilities and the kind-to-archetype map are assumed placeholders.

Runs hanger-fit's CMA-ES and random-search design loop (150 scores, 3 seeds) on five size M cases taken from hanger-sweep's corners: base tee, stiff (EA 800), soft (EA 40), soft and double weight, and soft, double weight on a 10 cm strip. The best arm half-width is 0.9 cm past the half shoulder for the tee and 1.5–1.8 cm for the stretchy cases, but slope, bow and end radius scatter between seeds at similar error, so only the width is identified. A tee-designed hanger scores 0.22 cm RMS on the tee and 1.06 cm on the soft heavy knit; one designed for the knit scores 0.52 and 0.76 cm. CMA-ES beats random search in all five cases. The narrow-strip case is poor for every hanger (best 0.99 cm). Fabric constants are swept corners, not measurements; 2D model, one size, shape error only.

hanger-avoidpaired's first next step: besides skipping garments near the other arm's pick in progress, also skip garments near its predicted next target. Paired on 1000 seeded loads (two arms, load 25), lookahead is measurably worse than avoiding the current pick in 14 of 18 settings and indistinguishable in the other 4, down to −11.3 garments an hour (aiming, 0.15 m, unlimited window) and about −1 with a 15 s window; the best case is pinching at 0.30 m, +0.40 (−0.03 to 0.84). A negative result. Stylised: clearance, station times and the pile are earlier placeholders; one predictor only.

Two arms emptying one hanger-pick pile, each waiting if its grasp is closer than a clearance to the other's, in the hanger-cell model: at 0.30 m aiming falls from 201 to 191 garments hung or folded per hour (5.7 dryers) and blind pinching from 179 to 151, since pinching keeps arms busier and they meet more (18 blocked picks per load against 10). A third arm adds only 3 to 5 per hour. Stylised: the clearance, the spacing time and all station times are assumed.

A half-lap hub joint with two M4 bolts lets every hanger-fit width up to the 25.8 cm XXL print on a 220 mm bed as one part printed twice. In a beam model with assumed PLA constants, the joint, not the arm, is then the weak point: safety factor 3.7–13 against 40–70 for the arm, set by bolt bearing on the half-depth boss, and the joint needs 4–9 mm of depth against the arm’s 0.8 mm. Printed pegs instead of bolts fail (safety factor 0.9–1.8 for 4 mm pegs). Stylised; nothing was printed or tested.

hanger-fit's drape uses round-number fabric constants. This sweep re-scores its width finding on a full factorial of 162 size M tees (bending rigidity 0.003–1 gf·cm²/cm, weight ×0.5–2, stretch stiffness 40–800 N/m, strip depth 10–30 cm) and one-at-a-time for the child and XXL tees. The best arm half width is the half shoulder + 1 cm in 151 of 162 and + 2 cm in 11, all with stretch stiffness 40 N/m and mostly double weight or a 10 cm strip (best error 0.9–1.7 cm against about 0.2 cm); child and XXL stay at + 1 cm throughout. Bending rigidity alone changes the best error by under 10% (median 0.23–0.25 cm over a 330-fold range). The only sourced input is a patent's range for knit cotton bending rigidity (0.002–0.050, from a search excerpt); stretch, weight and depth ranges are swept, not measured, and vary independently.

An independent 3D mass-spring sheet that wraps over a round-ended arm (cross-check of hanger-3d), draped on the hangers of hanger-fit with the same fabric constants and mass. It picks the same best arm width as the 2D chain (22.8 cm for a size M tee; error correlation 0.994 over a width sweep), with error 0.1–0.4 cm higher near the optimum, and adds the bend across the arm: 119 deg/cm over a 0.5 cm arm against 34 over a 2 cm one. It disagrees for a heavy knit on a thin arm (17.8 cm against 5.6 cm error) and is not grid-converged (1.19 against 0.62 cm at 1 and 0.5 cm spacing). Stylised, assumed fabric constants.

hanger-lookahead's first next step: yield to the other arm's predicted next pick only when the overlap it would cause is at least tau seconds. Paired on 1000 seeded loads (two arms, load 25, 54 settings), gating cuts the loss of ungated lookahead (aiming at 0.15 m: −11.3 an hour, −0.2 at 20 s) and finds one gain: pinching with 0.30 m clearance, +0.7 to +1.35 garments an hour over avoiding only the current pick at every threshold from 0 to 20 s (best +1.35, 0.93 to 1.80). Other settings show no gain whose interval excludes zero consistently. Stylised: clearance, station times and the pile are earlier placeholders; one predictor; the threshold is swept, not learned.