Wrapping a Target in a Swarm: SHARKS in Three Dimensions

Press Play. The dots start scattered through a cube; a dot turns blue once it sits inside the shell around the target, the dashed sphere. Drag the cube to look from any side. With the starting settings, more than two thirds of the 16 agents reach the shell by epoch 5 and all of them by epoch 20. Try Line through target with Turn r at 0°: the agents crowd along their starting line and never all settle. Set the turn to 20° and the same swarm is stable by epoch 6.

The Steps c : d setting sets how far an agent moves for each rule in one epoch: c units toward the shell and d units away from its neighbor. Compare 3 : 2 with 2 : 3. When the center step is the larger one, this swarm is stable by epoch 8; when the dispersion step is larger, agents that arrive late keep getting pushed back out by the agents already in the shell, and this run never settles. The magazine found the same direction: favoring the Center Rule by one step per epoch made swarms settle much faster.

Each epoch, every agent applies the Dispersion Rule and then the Center Rule, in a new random order. An agent moves only if the destination voxel (a unit cube of the field) is empty. The field wraps at its faces, the NetLogo 3D default. The thin vertical lines drop each agent to the target’s height, so you can read which agents fly above it and which below. The chart under the field tracks the share of agents inside δ ± 4 at each epoch for the current run.

The magazine gives the rules but not the arena. This field is a 33 × 33 × 33 wrapping cube (the size that fits the magazine’s averages best among sizes from 29 to 37), ε = 4, and steps of two and three units. The 2-D layouts keep their 2-D definitions in the target’s horizontal plane: the line starts 20 units to one side of the target, and the line through the target lies in that plane. With r = 0 and a line through the target, the Dispersion Rule’s compass bearing points along the line, so only the tilt can move an agent off it; with r = 20° the bearing itself leaves the line.

From above, the agent finds its nearest neighbor, faces the opposite way, and then turns a little more, r degrees clockwise. That extra turn is what makes the swarm circle the target instead of just spreading out. From the side, the agent levels its nose and then tips it down if it is above the target or up if it is below. The tilt pulls the swarm toward the target’s height a little at every step, so agents do not drift to the top or bottom of the sphere.

Algorithm 2 sets B_DEST = B_NN + 180 + r degrees, sets pitch to zero, and then pitches by −t above the target and +t otherwise. A step of length d therefore splits into d cos t horizontally and d sin t vertically, and the horizontal part splits again into a component straight away from the neighbor and d cos t sin r to the side. The Center Rule uses heading and pitch toward the target, so it moves along the radius. Only the Dispersion Rule carries an agent around the sphere, and only through r.

This demo runs 25 swarms of each size on your device and plots, for each size, the average share of agents in the shell as the epochs go by. All four sizes pass 68 percent at about the same time. The last few agents are the slow part, and they take longer in a bigger swarm, because a crowded shell blocks more moves and gives every newcomer more neighbors to push it away. The magazine describes the same pattern.

The magazine reports that every population reached 68 percent stability at about the same time and that 100 percent took longer as the swarm grew (6.6 epochs for four agents, 12.2 for 32). It explains the slower small swarms by sparse dispersion and the slower large ones by a crowded Center Rule. In this page’s sweep of all 1,792 settings (five runs each), the 68 percent mark falls between 1.8 and 2.2 epochs for every population, and complete stability takes 7.4, 8.7, 13.0, and 13.3 epochs for 4, 8, 16, and 32 agents. The gap between 95 and 100 percent opens only at 32 agents, where one or two stragglers are enough to stay below 100 percent.

Each pair of bars compares the article’s average number of epochs with this page’s. Lower is faster. The page agrees on every direction the article reports: boxes that start near the target settle fastest and boxes in a far corner slowest, a 20-degree turn beats no turn, a larger sphere settles faster, and a bigger step toward the target beats a bigger step away from the neighbor.

The article does not say how big its field was, so this page tried several sizes and kept the one whose averages came closest to the article’s. Matching after that kind of tuning shows that the rules can produce the article’s pattern; it does not prove the page copied the authors’ exact setup. Notice the rows where the bars still disagree, such as the line through the target and the tilt.

The page’s averages count only runs that reached complete stability within 400 epochs, which is how the magazine’s averages read; the third column shows how many did. In this sweep, 7,707 of 8,960 runs became stable. Most failures share one setting: with c = 2 and d = 3, only 1,159 of 2,240 runs settled, because an agent outside the shell whose nearest neighbor sits inside it moves out three units for every two it moves in. The magazine reports averages for that setting, so its swarms evidently settled; why they did is one of the open questions below.

• Two rules and a small tilt are enough for a leaderless swarm to surround a target in three dimensions as well as on the ground.
• Starting close to the target matters more than the exact starting pattern: boxes in the middle of the field settled in about two epochs, boxes in a corner in about 15.
• A 20-degree turn, a larger sphere, and a bigger step toward the target each made swarms settle faster.
• Because the drones need only the direction to their closest neighbor and to the target, they keep working if some drones are lost and they do not rely on GPS, which can be jammed or faked.

• Boxed starts at the center, faces, and edges of the cube settled in 1.8, 2.7, and 8.7 epochs on average, and cornered boxes in 14.7.
• Every swarm size reached 68 percent stability at about the same time; complete stability took 6.6 epochs with four agents and 12.2 with 32.
• A 20-degree turn averaged 7.9 epochs against 11.5 with no turn, and raising the ideal distance from 8 to 12 units cut the time from 13.2 to 6.2 epochs.
• Moving three units per epoch toward the shell averaged 5.2 epochs against 14.2 for two units, while three units of dispersion averaged 13.6 against 5.8 for two.

• Placement. Cornered boxes were slowest (14.7 epochs); random, linear, and collinear starts took 13 to 14; boxes on the edges, faces, and center took 8.7, 2.7, and 1.8. The authors conclude that the initial distance to the target matters more than the shape of the starting distribution.
• Population. All sizes reached 68 percent together; complete stability took 6.6 epochs for four agents and 12.2 for 32.
• Tilt and rotation. Three tilt settings stabilized in eight to ten epochs, slightly slower at larger tilts, but swarms without tilt did not rotate freely around the target. Rotation of 20° averaged 7.9 epochs against 11.5 for none, and collinear starts and small populations could fail to stabilize without rotation.
• Stability region. Raising the ideal distance from 8 to 12 units cut the average from 13.2 to 6.2 epochs.
• Rule weighting. Center steps of two and three units averaged 14.2 and 5.2 epochs; dispersion steps of two and three averaged 5.8 and 13.6.
• 2-D summary. In 2-D fields, smaller populations and larger ideal distances settled faster, boxed starts on the target performed best, and collinear starts held the most agents.

Drones that can surround a fire, an oil spill, or a damaged building without a pilot keep people out of danger. A swarm with no leader has no single drone an attacker can knock out to stop it, and a swarm that does not use GPS cannot be fooled by fake GPS signals. Moving from a flat field to the open air is what flying drones, and swimming or space robots, need.

The 3-D extension costs almost nothing: one comparison (above or below the target) and one pitch adjustment per step. That keeps SHARKS within reach of small, cheap vehicles while it inherits the protocol’s security properties, which come from what agents do not use: no positions of other agents, no messages, and no satellite positioning.

The article’s security case rests on minimal information per agent: bearings rather than coordinates, no communication after launch, and no dependence on GPS. The same lack of verification lets adversarial agents herd a 2-D swarm, as the authors’ 2019 work showed, so the 3-D results set a baseline for measuring attacks and defenses in the setting real aerial swarms occupy.

Notes from building this page

These notes interpret or extend the article and are flagged for the authors’ review.

• Unstated details and this page’s choices. Field: a 33 × 33 × 33 wrapping voxel cube with the target at the center, one agent per voxel. Rule order: Dispersion, then Center, in a random agent order each epoch. Shell half-width ε = 4. Steps of two and three units. Tilts of 0°, 10°, 20°, and 30°; populations of 4, 8, 16, and 32; boxes of 10 × 10 × 10 voxels placed at the center, on a face, on an edge, or in a corner; the line starts keep their 2-D definitions in the target’s horizontal plane. Runs that do not reach 100 percent within 400 epochs count as never stable and are left out of averages.
• How the field size was chosen. The page ran the full design (seven layouts, four populations, four tilts, two rotations, two distances, and two step sizes for each rule) for field widths of 29 to 37 voxels and ε of 3 to 5, and kept the pair with the smallest root-mean-square log ratio against the 17 reported means. A 33-voxel field with ε = 4 fit best (ε = 5 tied). Laying the line starts in the horizontal plane fit better than reading “above” as vertical. The 2018 conference paper implies a 35-unit 2-D field, close to this fit. Because of this fitting, agreement with the article’s averages is evidence that the rules can produce them, not a reproduction.
• What agrees. Directions for every reported factor; placement means within 1.5 epochs of the article for corner and edge boxes and for random and linear starts (14.1, 9.3, 11.9, and 13.1 against 14.7, 8.7, and 13 to 14); four agents at 7.4 epochs against 6.6 and 32 at 13.3 against 12.2; 68 percent stability at about the same epoch for every population; rotation, ideal-distance, and center-step means within 2 epochs.
• What does not. Collinear starts average 18.4 epochs here against 13 to 14. Larger tilts settle faster here (12.2, 12.4, 9.3, and 7.6 epochs for 0°, 10°, 20°, and 30°) where the article found them slightly slower, and untilted swarms here still rotate about the target (about 37° per 20 epochs against 45° at 10° tilt in one tracked run). The dispersion-step contrast is weaker here (9.1 and 12.0 against 5.8 and 13.6). With c = 2 and d = 3, about half the runs never settle.
• A balanced design behind the means. The two-level means for rotation, ideal distance, center step, and dispersion step all average 9.70 epochs, and the seven placement means average 9.77. That pattern is what a balanced factorial design produces, every factor’s marginal means averaging to the same grand mean. If the design was balanced, the four step combinations satisfy c2d2 + c2d3 = 28.4 and c2d2 + c3d2 = 11.6, so the slowest cell, c = 2 with d = 3, averaged between 16.8 and 27.2 epochs. This page’s value for that cell is 18.5, from the runs that settled.
• Where the swarm settles on the sphere. The tilt always points toward the target’s height, so settled swarms here gather in a band around the target’s horizontal plane rather than covering the whole sphere (mean height of 2.7 units at t = 10° against 4.2 units at t = 0° in one tracked run with δ = 12). The article’s word “ensphere” may describe the shell agents occupy more than an even spread over it; a coverage measure would settle which.
• The same symbol twice. The article uses d for the dispersion step and, in “when d is increased from 8 units to 12,” for a length the same paragraph calls the ideal distance. This page reads that sentence as the ideal distance δ.
• The 2-D check. With the third dimension switched off, this engine makes the same moves, in the same order, with the same random draws as the site’s 2-D SHARKS engine: 45,360 epoch-by-epoch comparisons over 378 runs, every layout, three populations, both step ratios, and three rotations. The 2-D engine reproduces the 2018 paper’s never-stable collinear starts without rotation and their rescue at r = 1°, so this engine does as well.
• Open questions for the authors. The field size and ε; the tilt angles; whether “above the target” in the line start meant height or the 2-D north direction; the population sizes; how many runs each mean averages and whether runs that never settled were excluded; and what kept c = 2, d = 3 swarms from stalling.

Try this in class

1. In the first demo, pick Box, center, Box, face, Box, edge, and Box, corner with 16 agents. Record the epoch at which each becomes stable, then measure on the drawing about how far each box starts from the target. Make a scatter plot of starting distance against epochs and describe the pattern in one sentence.
2. In the second demo, set the tilt to 30°. Use the readout to find how much of a three-unit step goes sideways and how much goes up or down. Check the numbers with cosine and sine on a calculator.
3. Run the third demo twice, once with Random and once with Box, corner. For each population, subtract the epochs to 68 percent from the epochs to 100 percent. Explain why the last third of a swarm can take longer than the first two thirds.