Distributed Predictive Flocking for a Swarm of Miniature Vibration-Driven Robots in a Confined Arena
Authors: Prajol Shrestha, Siddhant Mohapatra, Filip Novkoski, Julian Oelhaf, Nicolas Vandewalle, Ana-Sunčana Smith, Andreas Maier
Organizations: Pattern Recognition Lab, Department of Computer Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058, Erlangen, Germany · PULS, Institute for Theoretical Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058, Erlangen, Germany · GRASP, Institute of Physics B5a, University of Liège, B4000, Liège, Belgium · Group for Computational Life Sciences, Division of Physical Chemistry, Ruđer Bošković Institute, 10000, Zagreb, Croatia
Coordinating many miniature robots in a confined arena, with no leader, central planner, or global map, is a core challenge for swarm robotics. As the swarm grows denser, collision avoidance becomes the limiting factor, and the number of safely coexisting robots depends on the formulation of the inter-robot safety constraint. We present a distributed predictive flocking controller in which each robot solves its own nonlinear model predictive control (NMPC) problem, optimizing swarm cohesion and alignment costs, inter-robot and obstacle safety, and a task objective, while exchanging only predicted trajectories with local neighbors. Within this generalized controller architecture, solved in real time, we compare six barrier and control-barrier-function (CBF) safety formulations on a digital twin of an open-source vibration-driven bristlebot in a 0.64m2 arena. A look-ahead discrete-time CBF performs best overall, with the fewest collisions and the widest inter-robot margins across nearly every swarm size tested. We deploy it on phototaxis, phototactic orbiting, and cohesion without a stimulus tasks, using swarm sizes starting from ten robots and up to the geometric packing limit. The swarm holds well: collisions stay low in small swarms, rising steeply as the packing limit approaches, while coordination and cohesion degrade gently rather than collapsing. The swarm is found not to trade safety for order, keeping both across the confined arena, with local alignment high at every swarm size.
Figures & tables
Figure 1: (Left) Two elliptical robots with minimum safe distance Dsafe and semi-axes a ( 3cm ) and b ( 1.5cm ). (Right) The physical GRASPion open-source bristlebot hardware.
Figure 2: Per-robot control architecture for the DMPC swarm. Each robot runs three concurrent subsystems. The Perception block estimates the robot’s state comprising position px and py , orientation θ , and velocity vx and vy . The Planning and Control block uses this state to compute a look-ahead reference based on the active swarm behavior (phototaxis or phototactic orbital) and then solves the local NMPC problem to produce control commands, which are passed through a Calibration layer before reaching the Actuator. The Communication block handles all inter-robot data exchange, where solid arrows denote state signals and dotted arrows denote predicted trajectory exchanges. Incoming broadcasts are filtered by a Neighbor Filter (within communication range Rcomm and K -nearest selection) and stored in the robot’s neighbors buffer before being used by the NMPC solver.
Physical Robot Footprint
Safety Buffer
Swarm Size
Total Area
Occupancy
Total Area
Occupancy
Nb
( cm2 )
(%)
( cm2 )
(%)
1
14.14
0.22
38.48
0.60
10
141.40
2.21
384.85
6.01
25
353.50
5.52
962.11
15.03
50
707.00
11.05
1924.23
30.07
Table 1: Swarm size, physical robot footprint, and safety buffer occupancy in a 0.8m×0.8m ( 0.64m2 ) arena. The physical footprint uses Abot≈14.14cm2 and the safety buffer area is Abuffer≈38.48cm2 ( Dsafe=7cm ). The green row ( Nb=50 ) marks the representative swarm size used for the behavior figures. The yellow row ( Nb=100 ) marks the largest swarm size that leaves some headroom to move. The red row marks the geometric non-overlapping packing limit ( Nmax=137 ).
Figure 3: Safety-formulation comparison across the six variants (phototaxis, free space, 10-seed mean, 95% CI bands). (a) Solver feasibility, (b) inter-robot collisions (distinct contacts over 9000 steps, log scale), and (c) mean per-robot QP solve time, versus swarm size Nb . The dashed vertical line marks the geometric packing limit Nmax=137 .
Variant
Feas. (%)
Solve (ms)
Coll. Nb=25
Coll. Nb=100
Hard BF
89.70±0.03
2.79±0.01
67±8
90608±701
Hard Dt-CBF [ 33 ]
92.44±0.05
2.70±0.01
58±7
99704±869
Slack BF
100.00±0.00
3.11±0.04
5±2
116979±417
Slack Dt-CBF
100.00±0.00
3.77±0.01
2±1
49251±614
LA Slack Dt-CBF
100.00±0.00
3.74±0.02
1±1
36505±493
Slack Dt-HOCBF
100.00±0.00
3.78±0.01
2±2
40628±807
Table 2: Safety-formulation comparison (phototaxis, free space, 10-seed mean ±95% CI). Collisions are cumulative counts of distinct robot-robot contacts over 9000 steps, reported at a low density ( Nb=25 ) and at a high-density operating point Nb=100 . Feasibility and solve time are at Nb=25 . Bold marks the best value in each column.
Figure 4: Safety margins of the six safety formulations versus swarm size (phototaxis with two static obstacles, deployed closed loop with the reactive collision-response layer active, 10-seed mean, 95% CI bands). (a) Steady-state mean nearest-neighbor distance drr , center to center, dashed horizontal line at Dsafe=7 cm. (b) Mean robot-obstacle clearance dro , robot center to obstacle surface, dashed horizontal line at Dsafe,obs−rm=4 cm, since the obstacle barrier bounds the center-to-center distance. The gray band spans the footprint-contact range, which varies with orientation for elliptical robots ( 2b – 2a between two robots, b – a from a robot center to an obstacle surface). The vertical dashed line marks the geometric packing limit.
Figure 5: Collective behavior in obstacle-free space with the LA Slack Dt-CBF formulation, Nb=50 . (a) Phototaxis, where the swarm stays cohesive and heading-aligned while tracking the moving light source. (b) Phototaxis with the coupling costs and avoidance constraints removed, where the robots pile up at the light peak and collide. (c) Phototactic orbiting, where the swarm co-rotates along the light’s intensity level set. (d) Cohesion without a stimulus, where the Reynolds costs alone keep the group cohesive. Each column corresponds to a certain time t , and the orange field is the light source. Supplementary video 1 shows these four behaviors in the obstacle-free arena.
Figure 6: Local polar order Φloc (top row) and average nearest-neighbor distance (NND; bottom row) versus swarm size for the deployed LA Slack Dt-CBF, one column per collective behavior (10-seed mean with 95% CI bands, steady state over the last 300s ). The columns are phototaxis (a, d), phototactic orbiting (b, e), and cohesion without a stimulus (c, f). Each panel compares the obstacle-free arena, the few-obstacle arena ( 2 obstacles), and the cluttered arena ( 12 obstacles, positions randomized per seed). The dotted horizontal line marks the buffer distance of Dsafe=7cm , while the vertical dashed line marks the geometric packing limit Nmax=137 .
Figure 7: Distinct contacts versus swarm size for the deployed LA Slack Dt-CBF. (a) Phototaxis, (b) phototactic orbiting, and (c) cohesion without a stimulus. Each panel compares the obstacle-free arena, the few-obstacle arena ( 2 ), and the cluttered arena ( 12 , positions randomized per seed). The vertical axis is log-scaled. The dashed vertical line marks the packing limit Nmax=137 . The band around each curve is the 95% confidence interval over 10 seeds.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 8: Collective behaviors with two static obstacles (LA Slack Dt-CBF formulation, Nb=50 ), one behavior per row. (a) Phototaxis, (b) phototaxis with the Reynolds coupling removed, (c) phototactic orbiting, and (d) cohesion without a stimulus. Columns are snapshots at t=0 , 225 , 450 , 675 and 900s . The red disks are the static obstacles and the orange field is the light source. Supplementary video 2 shows these four behaviors in the two-obstacle arena.
Figure 9: Collective behaviors with twelve static obstacles (LA Slack Dt-CBF formulation, Nb=50 ), one behavior per row. (a) Phototaxis, (b) phototaxis with the Reynolds coupling removed, (c) phototactic orbiting, and (d) cohesion without a stimulus. Columns are snapshots at t=0 , 225 , 450 , 675 and 900s . The red disks are the static obstacles and the orange field is the light source. Supplementary video 3 shows these four behaviors in the cluttered arena.
Parameter
Value
Model and actuation bounds
Arena A
0.8m×0.8m ( 0.64m2 )
Position bounds px,py
[−0.4,0.4]m
Linear speed v
[0.01,0.15]m/s
Angular rate ω
[−3.463,1.432]rad/s
Robot semi-axes a,b
3cm,1.5cm
Appendix
Table 3: Complete controller, safety, environment, and simulation parameters used for all experiments. Robot-scale lengths are in cm, the arena and sensing ranges in m, and speeds in m/s; cost and slack weights are dimensionless. Values are those of the deployed LA Slack Dt-CBF; the cost weights, horizon, arena and platform limits are shared by every formulation, while α , L , γ1,γ2 and the slack weights are re-tuned per formulation (Section 3.1 ). The angular-rate bounds are asymmetric, reflecting the vibration-driven platform’s differing left- and right-turn authority.
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