Continuous-Time Motion Planning and Steering with Global Convergence

Mohsen Amiri; Sima Didari; Davoud Ataee Tarzanagh; Hankyu Moon; Yeongdae Kwon

Figure 8: Proof of Theorem 1 (ii) (drawn for

\ell<k

; otherwise Lemma 12 (c) adds an end correction). (a) The cheapest curve (black) with the margin raised by

\delta_{N}

(dotted circle) and its

N

-piece interpolant (blue), which stays within

\delta_{N}

of it (shaded) and so clears the obstacle grown by

\tilde{r}

(grey). (b) In the free control points (green: feasible set; grey: cost levels and infeasible grid points), a feasible ball (blue) around the interpolant (blue open dot) contains a grid point (black), so the cheapest feasible grid point (orange) costs no more, and GlobMP descends from it (small dots: iterates; orange open dot: their limit, on the dashed cost level). (c) The resulting cost bounds, lowest at the bottom (dots as in (b)).

Figure 9: Robot platform and operator station. The operator’s finger joints come from Manus gloves and the wrist poses from VIVE trackers against two base stations; retargeting turns both into absolute joint targets for the

38

arm and hand joints of a Dexmate Vega-1 carrying two XHAND1 hands, sent at

30

Hz. Head and wrist cameras and the fingertip force sensors record alongside. The close-up is the same hand photographed on the bench and cut out of its background; the five fingertip pads,

120

taxels each, are tinted amber and the leader points at one of them; they are the only tactile surfaces the hand has, the thumb’s included. The palm and the finger links sense nothing. Only the resulting trajectories are used in this paper; none of the operator hardware is present at inference.

Figure 10: Given modalities, at the instant the tissue box changes hands (one episode of the handover task,

27

s at

30

Hz). Prompt : the task string the policy is conditioned on, one per episode. Vision : three of the four RGB streams. Tactile : both hands at that same instant, five fingertip pads of

120

taxels each, every pad unrolled into its

10\times 12

measurement grid, one square per taxel, shown as its rise over the pre-contact baseline on a stretched colour scale. Force : the same fingertip sensors over the whole episode, every finger on its own, one panel per hand. Both scales are the sensor’s own counts, as measured: the hand reports integers and the vendor documents no conversion to newtons, so the counts carry no unit and the figure claims none. The load passes from the right hand to the left at the handover and falls when the box is placed.

Figure 11: GlobMP on the Georgia Tech Robotarium: twelve GRITSBots on Conveyor in the

3.2\times 2.0

m arena, at

t=25

s. The paths the robots drove, starts (circles), goals (squares), and the virtual obstacles drawn from the map. The circle magnifies the run’s closest pass:

6

cm between the robot’s body and a belt.

Method	SR $\uparrow$	Viol. $\downarrow$	Score $\uparrow$	Time (s) $\downarrow$	Gen. SR $\uparrow$	Gen. Viol. $\downarrow$	Gen. AS $\downarrow$	Gen. CS $\downarrow$
Diffuser	$0.000$	$44.600\pm 11.298$	$1.612\pm 0.004$	$0.256\pm 0.099$	$0.000$	$39.700\pm 18.815$	$0.0396\pm 0.0022$	$0.4281\pm 0.0427$
Classifier Guidance	$0.100$	$27.800\pm 13.934$	$1.612\pm 0.004$	$0.548\pm 0.091$	$0.300$	$9.000\pm 11.340$	$0.0400\pm 0.0022$	$0.4352\pm 0.0552$
Gradient Guidance	$0.200$	$13.000\pm 6.885$	$1.613\pm 0.005$	$2.663\pm 0.107$	$\mathbf{1.000}$	$\mathbf{0.000}\pm 0.000$	$0.0404\pm 0.0023$	$0.4471\pm 0.0460$
Projected Diffusion	$0.400$	$38.500\pm 45.247$	$1.314\pm 0.463$	$103.729\pm 26.086$	$\mathbf{1.000}$	$\mathbf{0.000}\pm 0.000$	$0.0026\pm 0.0002$	$0.0092\pm 0.0024$
Primal–Dual	$0.000$	$30.600\pm 12.492$	$0.111\pm 0.320$	$4.426\pm 0.118$	$0.800$	$0.300\pm 0.640$	$20.1566\pm 0.3115$	$1.8145\pm 0.0146$
Augmented Lagrangian	$0.500$	$9.600\pm 10.394$	$-0.025\pm 0.000$	$4.229\pm 0.212$	$0.000$	$9.900\pm 7.382$	$1.8568\pm 0.4291$	$1.1980\pm 0.0542$

Appendix

Table 1: Maze2D Broad , in the format of Giaretta et al. (2026) ’s Table 5. Mean

\pm

standard deviation. The four Gen. columns score the generated trajectory. Best value(s) in bold, Time excepted. All baselines are run here over

10

evaluation episodes, sampling from the same diffusion planner we train, so the values are measured rather than quoted; our rows are deterministic and report a single trajectory.

Method	SR $\uparrow$	Viol. $\downarrow$	Score $\uparrow$	Time (s) $\downarrow$	Gen. SR $\uparrow$	Gen. Viol. $\downarrow$	Gen. AS $\downarrow$	Gen. CS $\downarrow$
Diffuser	$0.000$	$94.100\pm 7.327$	$1.612\pm 0.004$	$0.260\pm 0.106$	$0.000$	$82.800\pm 16.726$	$0.0396\pm 0.0022$	$0.4281\pm 0.0427$
Classifier Guidance	$0.000$	$64.400\pm 14.908$	$1.619\pm 0.005$	$0.549\pm 0.104$	$0.000$	$27.700\pm 23.367$	$0.0402\pm 0.0023$	$0.4385\pm 0.0559$
Gradient Guidance	$0.100$	$21.700\pm 17.211$	$1.622\pm 0.006$	$2.607\pm 0.086$	$\mathbf{1.000}$	$\mathbf{0.000}\pm 0.000$	$0.0404\pm 0.0028$	$0.4475\pm 0.0535$
Projected Diffusion	$0.100$	$83.200\pm 158.026$	$1.057\pm 0.721$	$756.264\pm 166.998$	$0.800$	$4.800\pm 9.765$	$0.0145\pm 0.0225$	$0.0221\pm 0.0258$
Primal–Dual	$0.000$	$125.900\pm 114.102$	$0.103\pm 0.294$	$4.379\pm 0.138$	$0.000$	$11.400\pm 4.409$	$20.1512\pm 0.3103$	$1.8129\pm 0.0146$
Augmented Lagrangian	$0.000$	$222.200\pm 181.693$	$-0.025\pm 0.000$	$4.394\pm 0.204$	$0.000$	$159.300\pm 40.020$	$2.5444\pm 0.6313$	$1.2043\pm 0.0519$

Appendix

Table 2: Maze2D Narrow , the six-keep-out setting. Mean

\pm

standard deviation. The four Gen. columns score the generated trajectory. Best value(s) in bold, Time excepted. All baselines are run here over

10

evaluation episodes, sampling from the same diffusion planner we train, so the values are measured rather than quoted; our row is deterministic and reports a single trajectory.

Figure 12: Executed rollouts on Maze2D Broad , one panel per method, under the benchmark’s PD controller. Baseline panels show all ten evaluation runs behind Table 1 , unfiltered; GlobMP is deterministic and returns one. Stretches drawn dark and marked

\times

are where the robot is inside a keep-out, and the badge counts the runs that entered one, which is

10(1-

SR

)

of the table. Every baseline of the table appears, in the table’s order, with GlobMP’s exact

e{=}4

row. The last cell magnifies every stretch where GlobMP runs inside the enforced margin, the region marked by the dashed box on its own panel: the continuous plan (dashed) is held off the keep-out by the margin, and the executed rollout (solid) dips into the margin band but stays outside the keep-out, which is what the margin is for.

Figure 13: Executed rollouts on Maze2D Narrow , drawn as in Figure 12 , over the ten evaluation runs behind Table 2 . Only DiRecT and GlobMP keep every run outside the keep-outs; the keep-outs leave little of the corridor, and most baselines enter one on every run. So little is left that GlobMP’s trajectory is inside the enforced margin along most of the route, so the last cell magnifies its deepest pass rather than every stretch.

Figure 14: D3IL avoiding-v0 . (a) The environment’s own view; the test-time constraints have no physical counterpart, so they appear only in (b) . (b) The scene as GlobMP sees it, with all ten discs and both half-planes. (c) Safety rate against task success over

100

episodes, one point per method: a point at the lower right reaches the goal without avoiding, and only the upper right does both. Brackets as in Table 3 .

Figure 15: All

100

rollouts of every method on D3IL avoiding-v0 , from the common start (green), GlobMP last. A rollout is drawn red if it violates any of the twelve constraints under the benchmark’s own predicate and navy otherwise, so the fraction of navy is the safety rate in the panel title. The six pillars belong to the environment; the six lavender regions are imposed only at test time.

Method	SR $\uparrow$	TS $\uparrow$	Steps(Safe) $\downarrow$	Time (s) $\downarrow$
Diffuser (unconstrained)	0.01	0.89	$61.00\pm 0.00$	$0.09\pm 0.00$
Classifier Guidance	0.00	0.01	—	$0.17\pm 0.02$
Gradient Guidance	0.24	0.97	$62.96\pm 7.27$	$1.16\pm 0.07$
Projected Diffusion	0.45	0.79	$65.18\pm 8.09$	$7.16\pm 3.11$
$[$ Primal–Dual $]$	0.05	0.00	$100.00\pm 0.00$	$0.74\pm 0.02$
$[$ Augmented Lagrangian $]$	0.00	0.00	—	$0.79\pm 0.01$

Appendix

Table 3: D3IL avoiding-v0 , in the format of Giaretta et al. (2026) ’s Table 6,

100

episodes per method; mean

\pm

standard deviation over episodes. Every value is our own execution of the released code from a single shared checkpoint. Bracketed rows required a non-default --guidance-last-steps 50 to run at all: at the default both crash the simulator. Boldface marks the best safety rate and task success among the methods that enforce the constraints; the cost columns carry none. GlobMP is the projection step of DiRecT’s sampler (replanning every

4

steps,

28

CPU workers for

50

episodes at a time), on the same seed; over four seeds it reaches

0.865

and

0.97

.

Figure 16: Best-of-

64

success rate for

N_{a}\in\{3,6,\dots,21\}

agents on every cell of the suite, one row per map and one column per velocity cap: an instance counts as solved when any of the

64

candidates a baseline samples is collision-free. This is the SR reported in Tables 4 – 11 ; our method still returns one trajectory.

Figure 17: Success of a single returned plan for

N_{a}\in\{3,6,\dots,21\}

agents, one panel per map, averaged over the three velocity caps: the first candidate each baseline samples, and the one trajectory our method returns. MDOC and MMD-CBS store no per-candidate data and are omitted. Figure 16 gives the best-of-

64

counterpart.

		Low ( $v_{\max}=0.483$ )					Medium ( $v_{\max}=0.571$ )					High ( $v_{\max}=0.675$ )
$N_{a}$	Method	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$
3	Diff. Policy †	65.0	48.2 ±46.5	0.67 ±0.77	0.999 ±0.001	0.5	65.0	48.2 ±46.5	0.67 ±0.77	0.999 ±0.001	0.3	65.0	48.2 ±46.5	0.67 ±0.77	0.999 ±0.001	0.5
	CD †	100.0	1.1 ±4.4	0.14 ±0.28	0.517 ±0.149	1.0 ±0.1	100.0	7.1 ±20.2	0.14 ±0.28	0.517 ±0.149	1.1 ±0.1	100.0	20.5 ±26.1	0.14 ±0.28	0.517 ±0.149	0.5 ±0.0
	MMD-CBS †	100.0	70.0 ±45.8	—	0.946 ±0.071	3.1 ±2.5	100.0	70.0 ±45.8	—	0.946 ±0.071	3.1 ±2.5	100.0	70.0 ±45.8	—	0.946 ±0.071	3.1 ±2.5
	MDOC †	75.0	—	—	0.572 ±0.400	39.0 ±6.1	75.0	—	—	0.572 ±0.400	34.8 ±4.3	75.0	—	—	0.572 ±0.400	39.6 ±4.8
	PCD-LB	65.0	12.3 ±20.2	0.02 ±0.05	0.325 ±0.135	2.8 ±0.8	35.0	4.6 ±11.9	0.01 ±0.02	0.316 ±0.133	4.2 ±0.9	25.0	3.3 ±9.1	0.01 ±0.01	0.307 ±0.130	3.2 ±0.6

Appendix

Table 4: Multi-robot planning performance on Empty for

N_{a}\in\{3,6,\dots,21\}

under the Low, Medium and High velocity caps. SR and CS measure trajectory feasibility, Coll reports inter-robot collisions, DA adherence to the environment-specific motion pattern, and Time the planning time per configuration: for GlobMP (Ours) the median over the configurations with the standard deviation as subscript, for the baselines the mean, so the two are not directly comparable. † marks a method that applies no velocity limit, so its three blocks report the same run; MMD-CBS also returns one plan per configuration rather than

64

candidates. A dash under Coll marks a count that is not commensurable with the others, as MDOC and MMD-CBS count colliding time steps where every other row counts agent pairs with any collision; MDOC’s CS is omitted because its plans that do not cover every agent were counted as feasible. MMD-CBS plans under

60

s per configuration and GlobMP (Ours) under

300

s per team.

		Low ( $v_{\max}=0.477$ )					Medium ( $v_{\max}=0.567$ )					High ( $v_{\max}=0.675$ )
$N_{a}$	Method	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$
3	Diff. Policy †	30.0	13.0 ±29.6	0.35 ±0.64	0.998 ±0.001	0.3	30.0	13.0 ±29.6	0.35 ±0.64	0.998 ±0.001	0.3	30.0	13.0 ±29.6	0.35 ±0.64	0.998 ±0.001	0.3
	CD †	80.0	0.0 ±0.0	0.05 ±0.08	0.611 ±0.138	1.1 ±0.0	80.0	0.0 ±0.0	0.05 ±0.08	0.611 ±0.138	0.4 ±0.0	80.0	0.4 ±1.0	0.05 ±0.08	0.611 ±0.138	1.1 ±0.1
	MMD-CBS †	70.0	35.0 ±47.7	—	0.895 ±0.113	2.5 ±1.7	70.0	50.0 ±50.0	—	0.895 ±0.113	2.5 ±1.7	70.0	55.0 ±49.7	—	0.895 ±0.113	2.5 ±1.7
	MDOC †	10.0	—	—	0.090 ±0.269	33.4 ±4.5	10.0	—	—	0.090 ±0.269	33.6 ±4.4	10.0	—	—	0.090 ±0.269	33.5 ±4.6
	PCD-LB	40.0	5.2 ±10.1	0.01 ±0.03	0.361 ±0.114	5.2 ±1.6	35.0	2.2 ±3.6	0.01 ±0.03	0.349 ±0.112	4.4 ±0.6	20.0	2.1 ±6.1	0.01 ±0.03	0.336 ±0.106	3.6 ±0.4

Appendix

Table 5: Multi-robot planning performance on Basic for

N_{a}\in\{3,6,\dots,21\}

under the Low, Medium and High velocity caps. SR and CS measure trajectory feasibility, Coll reports inter-robot collisions, DA adherence to the environment-specific motion pattern, and Time the planning time per configuration: for GlobMP (Ours) the median over the configurations with the standard deviation as subscript, for the baselines the mean, so the two are not directly comparable. † marks a method that applies no velocity limit, so its three blocks report the same run; MMD-CBS also returns one plan per configuration rather than

64

candidates. A dash under Coll marks a count that is not commensurable with the others, as MDOC and MMD-CBS count colliding time steps where every other row counts agent pairs with any collision; MDOC’s CS is omitted because its plans that do not cover every agent were counted as feasible. MMD-CBS plans under

60

s per configuration and GlobMP (Ours) under

300

s per team.

		Low ( $v_{\max}=0.565$ )					Medium ( $v_{\max}=0.617$ )					High ( $v_{\max}=0.675$ )
$N_{a}$	Method	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$
3	Diff. Policy †	0.0	0.0 ±0.0	0.56 ±0.63	0.999 ±0.001	0.4	0.0	0.0 ±0.0	0.56 ±0.63	0.999 ±0.001	0.4	0.0	0.0 ±0.0	0.56 ±0.63	0.999 ±0.001	0.5
	CD †	5.0	0.0 ±0.0	0.14 ±0.20	0.674 ±0.096	1.1 ±0.1	5.0	0.0 ±0.0	0.14 ±0.20	0.674 ±0.096	1.1 ±0.0	5.0	0.0 ±0.0	0.14 ±0.20	0.674 ±0.096	1.1 ±0.0
	MMD-CBS †	0.0	0.0 ±0.0	—	—	0.8 ±0.5	0.0	0.0 ±0.0	—	—	0.8 ±0.5	0.0	0.0 ±0.0	—	—	0.8 ±0.5
	MDOC †	0.0	—	—	0.000 ±0.000	34.0 ±5.3	0.0	—	—	0.000 ±0.000	45.5 ±8.4	0.0	—	—	0.000 ±0.000	50.9 ±17.6
	PCD-LB	10.0	5.2 ±19.5	0.01 ±0.02	0.386 ±0.134	5.1 ±1.5	10.0	4.8 ±20.1	0.00 ±0.01	0.381 ±0.133	2.1 ±0.6	10.0	4.7 ±19.7	0.00 ±0.01	0.376 ±0.132	4.0 ±0.8

Appendix

Table 6: Multi-robot planning performance on Dense for

N_{a}\in\{3,6,\dots,21\}

under the Low, Medium and High velocity caps. SR and CS measure trajectory feasibility, Coll reports inter-robot collisions, DA adherence to the environment-specific motion pattern, and Time the planning time per configuration: for GlobMP (Ours) the median over the configurations with the standard deviation as subscript, for the baselines the mean, so the two are not directly comparable. † marks a method that applies no velocity limit, so its three blocks report the same run; MMD-CBS also returns one plan per configuration rather than

64

candidates. A dash under Coll marks a count that is not commensurable with the others, as MDOC and MMD-CBS count colliding time steps where every other row counts agent pairs with any collision; MDOC’s CS is omitted because its plans that do not cover every agent were counted as feasible. MMD-CBS plans under

60

s per configuration and GlobMP (Ours) under

300

s per team.

		Low ( $v_{\max}=1.21$ )					Medium ( $v_{\max}=1.46$ )					High ( $v_{\max}=1.76$ )
$N_{a}$	Method	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$
3	Diff. Policy †	5.0	4.9 ±21.5	0.37 ±0.53	0.016 ±0.070	0.6	5.0	4.9 ±21.5	0.37 ±0.53	0.016 ±0.070	0.6	5.0	4.9 ±21.5	0.37 ±0.53	0.016 ±0.070	0.6
	CD †	45.0	17.4 ±35.0	0.02 ±0.04	0.001 ±0.003	0.6 ±0.1	45.0	18.3 ±36.2	0.02 ±0.04	0.001 ±0.003	1.2 ±0.1	45.0	19.2 ±37.3	0.02 ±0.04	0.001 ±0.003	1.1 ±0.1
	MMD-CBS †	20.0	20.0 ±40.0	—	0.000 ±0.000	1.3 ±0.7	20.0	20.0 ±40.0	—	0.000 ±0.000	1.3 ±0.7	20.0	20.0 ±40.0	—	0.000 ±0.000	1.3 ±0.7
	MDOC †	5.0	—	—	0.000 ±0.000	40.1 ±7.0	5.0	—	—	0.000 ±0.000	36.4 ±2.6	5.0	—	—	0.000 ±0.000	36.3 ±2.8
	PCD-LB	70.0	36.2 ±42.5	0.01 ±0.03	0.004 ±0.016	3.0 ±0.1	65.0	37.1 ±43.6	0.01 ±0.02	0.004 ±0.016	3.1 ±0.1	65.0	36.8 ±43.5	0.01 ±0.02	0.004 ±0.017	3.1 ±0.1

Appendix

Table 7: Multi-robot planning performance on Conveyor for

N_{a}\in\{3,6,\dots,21\}

under the Low, Medium and High velocity caps. SR and CS measure trajectory feasibility, Coll reports inter-robot collisions, DA adherence to the environment-specific motion pattern, and Time the planning time per configuration: for GlobMP (Ours) the median over the configurations with the standard deviation as subscript, for the baselines the mean, so the two are not directly comparable. † marks a method that applies no velocity limit, so its three blocks report the same run; MMD-CBS also returns one plan per configuration rather than

64

candidates. A dash under Coll marks a count that is not commensurable with the others, as MDOC and MMD-CBS count colliding time steps where every other row counts agent pairs with any collision; MDOC’s CS is omitted because its plans that do not cover every agent were counted as feasible. MMD-CBS plans under

60

s per configuration and GlobMP (Ours) under

300

s per team. DA on this map scores an end-to-end traversal of one conveyor lane, which the uniformly drawn start and goal pairs almost never require; it is therefore below

0.07

for every method and does not separate them.

		Low ( $v_{\max}=0.647$ )					Medium ( $v_{\max}=0.783$ )					High ( $v_{\max}=0.878$ )
$N_{a}$	Method	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$
3	Diff. Policy †	10.0	10.0 ±30.0	0.15 ±0.35	0.567 ±0.298	0.3	10.0	10.0 ±30.0	0.15 ±0.35	0.567 ±0.298	0.3	10.0	10.0 ±30.0	0.15 ±0.35	0.567 ±0.298	0.3
	CD †	100.0	0.0 ±0.0	0.25 ±0.32	0.564 ±0.289	1.1 ±0.1	100.0	0.0 ±0.0	0.25 ±0.32	0.564 ±0.289	1.2 ±0.1	100.0	0.0 ±0.0	0.25 ±0.32	0.564 ±0.289	1.2 ±0.1
	MMD-CBS †	90.0	45.0 ±49.7	—	0.567 ±0.318	3.8 ±3.2	90.0	55.0 ±49.7	—	0.567 ±0.318	3.8 ±3.2	90.0	55.0 ±49.7	—	0.567 ±0.318	3.8 ±3.2
	MDOC †	15.0	—	—	0.083 ±0.233	41.3 ±5.6	15.0	—	—	0.083 ±0.233	37.2 ±3.4	15.0	—	—	0.083 ±0.233	36.8 ±3.3
	PCD-LB	0.0	0.0 ±0.0	0.02 ±0.02	0.565 ±0.293	3.5 ±0.4	0.0	0.0 ±0.0	0.05 ±0.11	0.564 ±0.292	3.0 ±0.2	0.0	0.0 ±0.0	0.05 ±0.13	0.564 ±0.292	3.0 ±0.1

Appendix

Table 8: Multi-robot planning performance on Highways for

N_{a}\in\{3,6,\dots,21\}

under the Low, Medium and High velocity caps. SR and CS measure trajectory feasibility, Coll reports inter-robot collisions, DA adherence to the environment-specific motion pattern, and Time the planning time per configuration: for GlobMP (Ours) the median over the configurations with the standard deviation as subscript, for the baselines the mean, so the two are not directly comparable. † marks a method that applies no velocity limit, so its three blocks report the same run; MMD-CBS also returns one plan per configuration rather than

64

candidates. A dash under Coll marks a count that is not commensurable with the others, as MDOC and MMD-CBS count colliding time steps where every other row counts agent pairs with any collision; MDOC’s CS is omitted because its plans that do not cover every agent were counted as feasible. MMD-CBS plans under

60

s per configuration and GlobMP (Ours) under

300

s per team.

		Low ( $v_{\max}=0.928$ )					Medium ( $v_{\max}=1.13$ )					High ( $v_{\max}=1.34$ )
$N_{a}$	Method	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$
3	Diff. Policy †	0.0	0.0 ±0.0	0.13 ±0.34	0.139 ±0.175	0.3	0.0	0.0 ±0.0	0.13 ±0.34	0.139 ±0.175	0.3	0.0	0.0 ±0.0	0.13 ±0.34	0.139 ±0.175	0.3
	CD †	45.0	5.2 ±21.7	0.11 ±0.32	0.000 ±0.001	0.5 ±0.0	45.0	11.9 ±24.6	0.11 ±0.32	0.000 ±0.001	1.0 ±0.2	45.0	22.2 ±34.5	0.11 ±0.32	0.000 ±0.001	1.2 ±0.0
	MMD-CBS †	30.0	25.0 ±43.3	—	0.000 ±0.000	2.2 ±1.5	30.0	30.0 ±45.8	—	0.000 ±0.000	2.2 ±1.5	30.0	30.0 ±45.8	—	0.000 ±0.000	2.2 ±1.5
	MDOC †	25.0	—	—	0.000 ±0.000	42.7 ±6.5	25.0	—	—	0.000 ±0.000	33.2 ±2.7	25.0	—	—	0.000 ±0.000	38.6 ±3.8
	PCD-LB	75.0	55.9 ±44.1	0.00 ±0.01	0.000 ±0.000	3.1 ±0.1	75.0	56.8 ±44.0	0.01 ±0.02	0.000 ±0.000	2.7 ±0.6	80.0	57.0 ±43.5	0.01 ±0.03	0.000 ±0.000	3.0 ±0.1

Appendix

Table 9: Multi-robot planning performance on Drop-Region for

N_{a}\in\{3,6,\dots,21\}

under the Low, Medium and High velocity caps. SR and CS measure trajectory feasibility, Coll reports inter-robot collisions, DA adherence to the environment-specific motion pattern, and Time the planning time per configuration: for GlobMP (Ours) the median over the configurations with the standard deviation as subscript, for the baselines the mean, so the two are not directly comparable. † marks a method that applies no velocity limit, so its three blocks report the same run; MMD-CBS also returns one plan per configuration rather than

64

candidates. A dash under Coll marks a count that is not commensurable with the others, as MDOC and MMD-CBS count colliding time steps where every other row counts agent pairs with any collision; MDOC’s CS is omitted because its plans that do not cover every agent were counted as feasible. MMD-CBS plans under

60

s per configuration and GlobMP (Ours) under

300

s per team.

		Low ( $v_{\max}=1.21$ )					Medium ( $v_{\max}=1.46$ )					High ( $v_{\max}=1.76$ )
$N_{a}$	Method	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$
3	Diff. Policy †	0.0	0.0 ±0.0	0.31 ±0.53	0.997 ±0.002	0.3	0.0	0.0 ±0.0	0.31 ±0.53	0.997 ±0.002	0.3	0.0	0.0 ±0.0	0.31 ±0.53	0.997 ±0.002	0.3
	CD †	85.0	12.5 ±20.0	0.12 ±0.26	0.683 ±0.079	1.2 ±0.0	85.0	16.3 ±21.3	0.12 ±0.26	0.683 ±0.079	1.1 ±0.0	85.0	18.0 ±21.5	0.12 ±0.26	0.683 ±0.079	1.2 ±0.0
	MMD-CBS †	45.0	45.0 ±49.7	—	0.890 ±0.068	5.7 ±6.5	45.0	45.0 ±49.7	—	0.890 ±0.068	5.7 ±6.5	45.0	45.0 ±49.7	—	0.890 ±0.068	5.7 ±6.5
	MDOC †	10.0	—	—	0.072 ±0.215	36.1 ±6.6	10.0	—	—	0.072 ±0.215	36.4 ±2.9	10.0	—	—	0.072 ±0.215	37.0 ±3.6
	PCD-LB	10.0	0.5 ±1.7	0.02 ±0.06	—	3.0 ±0.0	10.0	0.5 ±2.1	0.01 ±0.05	—	3.1 ±0.1	20.0	1.5 ±3.8	0.01 ±0.03	—	3.1 ±0.1

Appendix

Table 10: Multi-robot planning performance on Corridor-Bay for

N_{a}\in\{3,6,\dots,21\}

under the Low, Medium and High velocity caps. SR and CS measure trajectory feasibility, Coll reports inter-robot collisions, DA adherence to the environment-specific motion pattern, and Time the planning time per configuration: for GlobMP (Ours) the median over the configurations with the standard deviation as subscript, for the baselines the mean, so the two are not directly comparable. † marks a method that applies no velocity limit, so its three blocks report the same run; MMD-CBS also returns one plan per configuration rather than

64

candidates. A dash under Coll marks a count that is not commensurable with the others, as MDOC and MMD-CBS count colliding time steps where every other row counts agent pairs with any collision; MDOC’s CS is omitted because its plans that do not cover every agent were counted as feasible. MMD-CBS plans under

60

s per configuration and GlobMP (Ours) under

300

s per team.

		Low ( $v_{\max}=0.928$ )					Medium ( $v_{\max}=1.13$ )					High ( $v_{\max}=1.34$ )
$N_{a}$	Method	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$	SR $\uparrow$	CS $\uparrow$	Coll $\downarrow$	DA $\uparrow$	Time $\downarrow$
3	Diff. Policy †	0.0	0.0 ±0.0	0.30 ±0.55	0.985 ±0.007	0.6	0.0	0.0 ±0.0	0.30 ±0.55	0.985 ±0.007	0.5	0.0	0.0 ±0.0	0.30 ±0.55	0.985 ±0.007	0.5
	CD †	20.0	5.6 ±15.6	0.02 ±0.05	0.662 ±0.070	1.1 ±0.0	20.0	9.3 ±25.5	0.02 ±0.05	0.662 ±0.070	1.1 ±0.1	20.0	10.1 ±27.3	0.02 ±0.05	0.662 ±0.070	1.1 ±0.0
	MMD-CBS †	10.0	10.0 ±30.0	—	0.887 ±0.049	0.7 ±0.4	10.0	10.0 ±30.0	—	0.887 ±0.049	0.7 ±0.4	10.0	10.0 ±30.0	—	0.887 ±0.049	0.7 ±0.4
	MDOC †	0.0	—	—	0.000 ±0.000	37.9 ±2.4	0.0	—	—	0.000 ±0.000	36.2 ±2.5	0.0	—	—	0.000 ±0.000	35.4 ±2.6
	PCD-LB	55.0	16.6 ±30.6	0.05 ±0.17	—	2.7 ±0.2	45.0	15.6 ±30.8	0.05 ±0.15	—	1.5 ±0.1	45.0	16.3 ±30.6	0.04 ±0.14	—	1.5 ±0.1

Appendix

Table 11: Multi-robot planning performance on Cross-Room for

N_{a}\in\{3,6,\dots,21\}

under the Low, Medium and High velocity caps. SR and CS measure trajectory feasibility, Coll reports inter-robot collisions, DA adherence to the environment-specific motion pattern, and Time the planning time per configuration: for GlobMP (Ours) the median over the configurations with the standard deviation as subscript, for the baselines the mean, so the two are not directly comparable. † marks a method that applies no velocity limit, so its three blocks report the same run; MMD-CBS also returns one plan per configuration rather than

64

candidates. A dash under Coll marks a count that is not commensurable with the others, as MDOC and MMD-CBS count colliding time steps where every other row counts agent pairs with any collision; MDOC’s CS is omitted because its plans that do not cover every agent were counted as feasible. MMD-CBS plans under

60

s per configuration and GlobMP (Ours) under

300

s per team.

Experiment	Section	degree $d$	$k$	$\ell$
Synthetic study	§ 4.1	$9$	$4$	$4$
Multi-robot suite	§ A4.5	$9$	$2$	$1$
Robotarium	§ 4.2	$12$	$2$	$1$
Maze2D	§ A4.3	$9$	$2$	$2$

Appendix

Table 12: Curve class of each planning experiment.

Figure 18: Hardware execution on Empty : every method except PCD-LB, twelve GRITSBots on the Georgia Tech Robotarium. The tracking controller and the virtual obstacles are those of Appendix A4.1 . Each panel reports one execution of the ring instance by the method named above it: a single overhead frame, with the trajectories recovered from the video drawn over it, each marked by a circle at its start and a square at its terminal state. A robot is counted as being in collision when its footprint overlaps an obstacle region. The frame shown is the midpoint of the tracked window for an execution in which no robot overlaps a region at any time, and otherwise the frame at which the number of overlapping robots is maximal; only the midpoints are therefore proportional to the duration of an execution. Empty carries no obstacles, so no execution can overlap a region and the panels differ only in the routing of the exchange, with midpoints at

25

s for GlobMP,

27

s for PCD-SHD,

28

s for Final Proj.,

29

s for DiRecT,

34

s for CD and

57

s for MMD-CBS.

Figure 19: Hardware execution on Drop-Region : the same methods, with the protocol of Figure 18 . Three of the six methods complete the exchange with no footprint overlap: MMD-CBS (midpoint frame at

61

s), DiRecT (

31

s) and GlobMP (

31

s). For the remaining three the panel shows the frame of maximal overlap: two of the twelve robots overlap a keep-out region under Final Proj. (

40

s), three under CD (

43

s) and five under PCD-SHD (

23

s). PCD-SHD is additionally the only execution on this map whose robots do not reach their goals; its terminal states lie well away from the goal ring. This map discriminates least among the four maps with obstacles, and the discrimination that remains lies in when the overlaps occur rather than in how many robots they involve.

Figure 20: Hardware execution on Conveyor : the same methods, with the protocol of Figure 18 . The three belts sit in the middle of the arena and leave two gaps between them, so the exchange requires an ordering of the team as well as a route; the executions differ principally in that ordering. GlobMP completes the instance with no footprint overlap (midpoint frame at

37

s). PCD-SHD and DiRecT each leave five of the twelve robots overlapping a belt at their frames of maximal overlap,

44

s and

65

s respectively, the latter being the latest frame on this map. CD and Final Proj. leave all twelve overlapping, at

40

s and

48

s: both traverse the belts as a single block, so the entire team is inside a region at one instant. MMD-CBS returned no plan on this map; its robots therefore remain on the start ring, and its panel is marked rather than scored, since counting overlaps for a team that never moved would record a success where no trajectory was produced.

Figure 21: Hardware execution on Corridor-Bay : the same methods, with the protocol of Figure 18 . As on Drop-Region , three methods complete the exchange with no footprint overlap: PCD-SHD (midpoint frame at

45

s), DiRecT (

32

s) and GlobMP (

26

s). The remaining three overlap the keep-out regions at their frames of maximal overlap: three of the twelve robots under MMD-CBS (

42

s), four under Final Proj. (

78

s), the latest frame of maximal overlap in this gallery, and eight under CD (

49

s). Final Proj. remains clear for the first minute and overlaps only as the team disperses towards its goals, whereas CD overlaps from early in the execution. MMD-CBS, which returns no plan on two of the other maps, produces one here and brings nine of the twelve robots through without overlap.

Figure 22: Hardware execution on Cross-Room : the same methods, with the protocol of Figure 18 . The cross occupies the centre of the arena and leaves four corner regions connected only along the walls, so the exchange must be routed around the arms; a team that advances directly towards its goals enters one of them. GlobMP is the only method that completes the instance with no footprint overlap (midpoint frame at

26

s). At their frames of maximal overlap, six of the twelve robots overlap the cross under DiRecT (

37

s) and seven under PCD-SHD (

41

s); both route part of the team around the arms and lose the remainder. CD and Final Proj. leave all twelve overlapping, at

38

s and

34

s. MMD-CBS returned no plan on this map as well, so its panel again shows the untouched start ring and is marked rather than scored. With Conveyor , this is the map on which the margin between GlobMP and the baselines is largest under hardware execution.

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