Dispersive Tree Agreements for Multi-Robot Coordination

Shashank A. Deshpande; Blake T. Edwards; Jonathan P. How

cs.ROOct 22, 2026

Dispersive Tree Agreements for Multi-Robot Coordination

Authors: Shashank A. Deshpande, Blake T. Edwards, Jonathan P. How

Organizations: Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract

Multi-robot coordination of nonlinear systems is naturally formulated as joint-space optimal control subject to constraints over groups of robots. Existing methods that approach joint optimality remain largely centralized, offline, and confined to collision-free motion planning, while decentralized, real-time counterparts target safe navigation without pursuing the joint optimum. This paper develops a decentralized, real-time planner that approaches the joint optimum as an anytime process. We extend a recent single-robot framework that searches dispersive trees of forward-propagated commands for differentially flat systems to coordinate a multi-robot team near-optimally. We first show that the inter-robot coupling can be effectively replaced by some local constraints, under which locally grown dispersive trees contain a near-optimal agreement. It is this structure that the robots pursue by decentralized column generation, negotiating rents on the shared space that guide the search of each robot's tree. The negotiation settles on an agreement whose near-optimality is certified at quiescence. Our protocol, WarpWeaver, implements this negotiation as an asynchronous real-time loop over a receding horizon, and demonstrates effective closure to quiesced agreements even when real-time deadlines cut the negotiation short. In simulation, WarpWeaver reduces total travel time by up to 22% against centralized offline planners on the Dynobench suite and by up to 37% against a decentralized real-time baseline, and nearly doubles the success rate of a task-specific planner on graph connectivity. Hardware experiments on six robots planning onboard demonstrate real-world implementability.

Figures & tables

Fig. 1: (a) Iterated Best Response, $F={\color[rgb]{0.168,0.4297,0.5508}7.5}\;{+}\;{\color[rgb]{0.707,0.3125,0.1719}10.0}=17.5$ s, vs. (b) Dispersive Tree Agreement, $F={\color[rgb]{0.168,0.4297,0.5508}8.0}\;{+}\;{\color[rgb]{0.707,0.3125,0.1719}5.5}=13.5$ s, where the total travel time $F$ sums the arrival times of the blue and red robots.

Fig. 2: Decentralized column generation. The dual of the restricted master over the pool

Q

sets common prices, under which each agent searches its own dispersive tree. Sequences that improve the pool return as new columns until the agreement is certified at quiescence.

Fig. 3: The WarpWeaver protocol ( Algorithm 1 ), visualized as a flowchart at the onboard computer of agent

i

. Each agent asynchronously runs an identical loop onboard.

Panel of Figure 4 — Algorithm 1 WarpWeaver at agent $i$

Platform	$f_{i}$ , $U_{i}$	$E_{i}$	$R_{i}$	$B_{i}$ (m)	$\mathcal{A}$	Search Parameters
Unicycle	$\dot{p}=ve_{\theta}$ , $\dot{\theta}=\omega$ , $\|v\|,\|\omega\|\leq 0.5$	–	–	Box $0.5\times 0.25$	Constant $(v,\omega)$ on a lattice $(v,\omega)\in G_{8}\bigl[-\tfrac{1}{2},\tfrac{1}{2}\bigr]^{2}$	$\Delta=0.5,\ T=10$ , $\Delta_{s}=0.5,\ s_{i}=0.08$ , $\varepsilon_{g}=0.5$ , $S=0.6,\ n_{s}=5$
Double Int. (3D)	$\ddot{p}=a$ , $\\|a\\|_{\infty}\leq 2$	$\\|v\\|_{\infty}\leq 0.5$	$\\|v\\|_{\infty}\leq 0.17$	Ball $0.1$	Constant $a$ on a lattice of the admissible box $a\in G_{4}\bigl(J(x)\bigr)$ , with $J(x)=[-2,2]^{3}\cap\bigl([-\tfrac{1}{2},\tfrac{1}{2}]^{3}-v\bigr)/\Delta$	$\Delta=0.5,\ T=10$ , $\Delta_{s}=0.5,\ s_{i}=0.25$ , $\varepsilon_{g}=0.5$ , $S=0.25,\ n_{s}=5$
Triple Int. (3D)	$\dddot{p}=j$ , $\\|j\\|\leq 25$	$\\|v\\|\leq 4$ , $\\|a\\|\leq 8$	$\\|v\\|_{\infty}\leq 0.3$ , $\\|a\\|_{\infty}\leq 0.6$	Ball $0.15$	Admissible jerk profiles to diverse reference velocities $j\in\bigl\{j^{1}_{\bar{v}},\,j^{2}_{\bar{v}}:\bar{v}\in V(x)\bigr\}$	$\Delta=0.5,\ T=8$ , $\Delta_{s}=0.5,\ s_{i}=2$ , $\varepsilon_{g}=0.75$ , $S=0.35,\ n_{s}=10$
X500	$\dot{p}=v$ , $m\dot{v}=c\,b_{3}(q)-mge_{3}$ , $\dot{q}=\tfrac{1}{2}q\otimes\omega$ , $J\dot{\omega}=M-\omega\times J\omega$ , $(c,M)=B\eta$ , $t_{m}\dot{\eta}=u-\eta$ , $u\in[0.19,8.55]^{4}$	$\\|v\\|\leq 4$ , $\|\omega_{1,2}\|\leq 3.84$ , $\|\omega_{3}\|\leq 3.49$ , $\angle(b_{3},e_{3})\leq 45^{\circ}$	$\\|v\\|_{\infty}\leq 0.5$ , $\\|\omega\\|_{\infty}\leq 0.5$	Ball $0.1$	Smooth acceleration references on a lattice, tracked by the geometric controller of [ 17 ] $a\in G_{5}[-8,8]^{2}\times\{-8,-4,0,1.85,3.7\}$ , $c_{-}\leq\\|a+ge_{3}\\|\leq c_{+}$	$\Delta=0.25,\ T=6$ , $\Delta_{s}=0.5,\ s_{i}=2$ , $\varepsilon_{g}=0.75$ , $S=0.6,\ n_{s}=25$
Single Int. (2D)	$\dot{p}=u$ , $\\|u\\|\leq 1.5$	–	–	Disk $0.36$	Constant $u$ on a polar lattice $u\in\{\sigma e_{2\pi k/16}:\sigma\in\{0.5,1,1.5\},$ $k<16\}$	$\Delta=1,\ T=10$ , $\Delta_{s}=1,\ s_{i}=0.1$ , $\varepsilon_{g}=1.5$ , $S=0.75,\ n_{s}=10$

TABLE I: Platforms, command sets, and search parameters. Commands are held over windows of length

\Delta

(s) [ 17 ] , and each set also contains a hold and, for the integrators, a brake toward rest.

G_{n}[l,u]

: the

n

-level grid on

[l,u]

;

e_{\theta}=(\cos\theta,\sin\theta)

. Triple integrator at

x=(p,v,a)

j^{1}_{\bar{v}}=-a/\Delta\pm 4\lambda_{1}(\bar{v}-v-\Delta a/2)/\Delta^{2}

, with

+

on the first half window, and

j^{2}_{\bar{v}}=2\lambda_{2}(\bar{v}-v-\Delta a)/\Delta^{2}

, with

\lambda_{1},\lambda_{2}\leq 1

maximal within the bounds;

V(x)=\{\sigma(\cos\phi\,e_{\psi+\theta},\sin\phi)\}

\sigma\in\{\frac{4}{3},\frac{8}{3},4\}

\theta\in\{0,\pm 5^{\circ},\pm 10^{\circ},\pm 20^{\circ}\}

\phi\in\{0,\pm 20^{\circ}\}

\psi

the heading of

v

. X500: attitude

q

, body axis

b_{3}

, body rate

\omega

, rotor thrusts

\eta

(N), mixer

B

, motor time constant

t_{m}

T

: horizon in windows;

\Delta_{s}

(s): stride;

s_{i}

: pruning radius;

\varepsilon_{g}

: goal tolerance;

S

(m): cell diameter;

n_{s}

: substeps per window.

Panel of Figure 6 — Fig. 4: WarpWeaver in simulation. (a) Circle-10, unicycles. (b) Forest-10, 3D double integrators. (c) Ten triple integrators on a forest map of EGO-Swarm. (d) Connectivity maintenance in the forest world of Realm, with the live links at three instants. In (a) and (d), each robot has its own colour and is drawn at three instants, later ones more opaque. In (b) and (c), trajectories and bodies are coloured by time from start (circle) to goal (star).

	Instance	$N$	Warp- Weaver	Quie- sced	db-CBS	db-ECBS	db-PIBT	db-LaCAM @60 s
Unicycle	swap2	$2$	$\mathbf{11.0}$	$11.0$	$12.0$	$14.4$	$16.8$	$15.6$
	swap4	$4$	$\mathbf{24.0}$	$22.5$	$27.6$	$27.1$	–	$72.0$
	atgoal	$2$	$\mathbf{6.5}$	$7.0$	$8.4$	$8.3$	$25.2$	$16.8$
	circle6	$6$	$84.5$	$83.5$	$90.0$	$\mathbf{82.9}$	$97.2$	$96.0$
	circle8	$8$	$\mathbf{113.0}$	$113.0$	$121.2$	$117.8$	$141.6$	$136.8$
	circle10	$10$	$\mathbf{140.5}$	$139.5$	$150.0$	$156.3$	$182.4$	$196.8$

TABLE II: Multi-robot Dynobench: total travel time (s) summed over agents, median. Quiesced: WarpWeaver with the world paused until

\varepsilon_{q}

-quiescence or for at most

5

s ( Section VI-C ),

\varepsilon_{q}

-quiescent at more than

84\%

of the iterations of every run, median over five trials.

Map	Planner	Success	$F$ (min–max)
Forest 1	WarpWeaver, triple int.	$\mathbf{5/5}$	$\mathbf{91.5}$ (91.0–94.5)
	WarpWeaver, X500	$\mathbf{5/5}$	$97.5$ (97.5–101.0)
	EGO-Swarm	$4/5$	$145.7$ (142.4–148.5)
Forest 2	WarpWeaver, triple int.	$\mathbf{5/5}$	$\mathbf{92.5}$ (91.0–93.0)
	WarpWeaver, X500	$\mathbf{5/5}$	$99.0$ (98.0–102.0)
	EGO-Swarm	$\mathbf{5/5}$	$145.9$ (136.0–154.0)

TABLE III: Ten drones on the EGO-Swarm forest maps.

F

: total travel time (s) summed over agents, median (min–max) over successful trials.

	WarpWeaver		db-LaCAM
$N$	Total	per robot	Total	per robot
$10$	$167.5$ ( $167.0$ – $168.5$ )	$16.8$	$217.2$	$21.7$
$20$	$435.5$ ( $432.5$ – $438.0$ )	$21.8$	$522.0$	$26.1$
$30$	$649.5$ ( $645.0$ – $654.0$ )	$21.7$	$884.4$	$29.5$
$40$	$910.5$ ( $904.0$ – $913.0$ )	$22.8$	$1400.4$	$35.0$
$50$	$1126.5$ ( $1107.5$ – $1165.5$ )	$22.5$	$1470.0$	$29.4$

TABLE IV: Scaling tests, unicycles in a

20\times 20

m arena, five trials per

N

, WarpWeaver in real time (emulated). Total travel time summed over agents and per robot (s), median (min–max) for WarpWeaver and the median best solution of db-LaCAM within

300

World	Planner	Success	Time (s)	Conn.	Path
Clutter	WarpWeaver	$\mathbf{5/5}$	$\mathbf{23.9}$ (19.7–39.1)	$1.00$	$122$
Clutter	Realm	$\mathbf{5/5}$	$58.3$ (31.0–109.2)	$1.00$	$193$
Garage	WarpWeaver	$\mathbf{5/5}$	$\mathbf{43.8}$ (25.8–54.0)	$1.00$	$234$
Garage	Realm	$3/5$	$61.9$ (55.3–85.5)	$1.00$	$226$
Forest	WarpWeaver	$\mathbf{5/5}$	$\mathbf{54.6}$ (45.3–73.8)	$1.00$	$273$
Forest	Realm	$0/5$	–	$1.00$	–

TABLE V: Connectivity maintenance against Realm, four robots. Time: arrival of the last robot (s), median (min–max); Conn.: fraction of time connected, worst task; Path: summed path length (m), median.

	Quiesc.			$t_{q}$ , $\varepsilon_{q}=1$			$F$
$N$	RT	$\varepsilon_{q}=0$	$\varepsilon_{q}=1$	p50	p90	p99	RT	$\varepsilon_{q}=0$	$\varepsilon_{q}=1$
$10$	$70\%$	$98\%$	$100\%$	$0.11$	$0.31$	$1.26$	$167.5$	$167.5$	$167.0$
$20$	$16\%$	$58\%$	$99\%$	$1.32$	$2.41$	$>5$	$435.5$	$431.5$	$435.0$
$30$	$6\%$	$33\%$	$93\%$	$1.48$	$4.63$	$>5$	$649.5$	$643.0$	$658.0$
$40$	$1\%$	$7\%$	$60\%$	$3.56$	$>5$	$>5$	$910.5$	$899.5$	$898.5$
$50$	$0\%$	$2\%$	$7\%$	$>5$	$>5$	$>5$	$1126.5$	$1072.0$	$1065.5$

TABLE VI: Quiescence in real time and with the world paused (cap

5

s, one trial per

N

). Quiesc.: iterations that end quiescent, in real time (median over the trials of Table IV ) and within the cap at

\varepsilon_{q}=0

and

\varepsilon_{q}=1

;

t_{q}

: time to quiescence at

\varepsilon_{q}=1

(s), capped iterations counted as

>5

;

F

: total travel time (s) in real time (median, Table IV ) and with up to

5

s of negotiation per iteration.

Panel of Figure 12 — Fig. 5: Six robots, one take per task. Left: executed trajectories from motion capture, each robot in its own colour, with the robots at the start, midway, and at the last arrival, later instants more opaque. Right: the camera frames overlaid, later ones more opaque. (a) Swap around four pillars. (b) Tethered swap, with the tied pairs joined by dashed lines. (c) Doorway swap.

Task	Success	Last arrival	Total	Sep.	Tether
Swap	$3/3$	$20.2$ ( $20.0$ – $20.8$ )	$108.1$	$0.28$	–
Tethered swap	$3/3$	$30.4$ ( $24.5$ – $31.9$ )	$125.4$	$0.28$	$2.37$
Doorway swap	$3/3$	$33.6$ ( $25.3$ – $35.6$ )	$162.1$	$0.12$	–

TABLE VII: Real-robot experiments, six robots, three consecutive takes per task. Time: arrival of the last robot (s) and total travel time summed over robots (s), median (min–max); Sep.: smallest distance between two robots minus both clearance radii (m); Tether: largest distance within a tied pair (m), against

L=2.4

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