Pattern Formation for Asynchronous Robots without Agreement in Chirality

This paper presents a deterministic algorithm for forming a given asymmetric pattern in finite time by a set of autonomous, homogeneous, oblivious mobile robots under the CORDA model. The robots are represented as points on the 2D plane. There is no explicit communication between the robots. The robots coordinate among themselves by observing the positions of the other robots on the plane. Initially all the robots are assumed to be stationary. The robots have local coordinate systems defined by Sense of Direction (SoD), orientation or chirality and scale. Initially the robots are in asymmetric configuration. We show that these robots can form any given asymmetric pattern in finite time.

💡 Research Summary

The paper addresses the problem of forming an arbitrary asymmetric pattern with a swarm of autonomous, homogeneous, oblivious mobile robots operating under the CORDA model, which captures fully asynchronous behavior. Each robot is a point in the plane, has no unique identifier, no explicit communication, and possesses only a local coordinate system defined by its own origin, sense of direction (SoD), chirality (orientation), and unit distance. The robots have unlimited visibility and execute the classic Look‑Compute‑Move cycle asynchronously; movements are not instantaneous and may be observed by other robots while in transit.

The authors assume that the initial configuration of the robots is asymmetric, meaning that no line of symmetry exists that would split the set into mirror images. The target pattern is also asymmetric and is given as a set of n points in an arbitrary global coordinate system. The goal is to move the robots so that each robot occupies a distinct point of the pattern, achieving the configuration in finite time without collisions.

The contribution consists of a deterministic distributed algorithm that works without any common sense of direction or common chirality, thereby removing two major restrictions present in earlier works. Moreover, the algorithm eliminates the need for special conditions such as “the leader must lie on the smallest enclosing circle (SEC)” or “the number of robots inside the SEC must not exceed the number on the SEC,” which were required in prior solutions.

The algorithm is built from several sub‑procedures:

1. AgreementPattern() – Computes the SEC of the pattern, selects a point on the SEC as the leader of the pattern (pₗ), and chooses a second SEC point (p′ₗ) to define the positive X‑axis (vector c_P pₗ) and the positive Y‑axis (perpendicular direction determined by p′ₗ). The radius of the SEC becomes the common unit distance. Lemmas 1‑3 prove that this choice is unique and thus provides a globally consistent reference for all robots.

2. AgreementCoordinateSystem() – Computes the SEC of the robot set, selects the first robot on that SEC (rₗ) as the leader, and aligns the robot coordinate system with the pattern’s reference: the robot leader coincides with pₗ, the common origin is the SEC center, and the axes and unit distance are identical to those defined for the pattern. Lemmas 4‑5 guarantee that all robots agree on the same origin, axes, and scale despite having no shared global frame.

3. MoveRadiallyOut(rᵢ) – After establishing a common origin O, the algorithm identifies the pattern point p₁ nearest to O and the robot r₁ nearest to O. It defines d = max(|Or₁|, |Op₁|) and constructs a circle C(O, d + ε) where ε is a small positive constant. All robots that lie inside or on the circle of radius d are moved radially outward to the boundary of C(O, d + ε). The robot closest to this new boundary is selected (ties broken by the predetermined ordering). The movement is designed so that the destination point is unoccupied and the robot stays nearest to the boundary during motion, ensuring collision‑free behavior (Lemma 6).

4. MoveToDestination(rᵢ) – Handles the remaining robots and pattern points. It builds two sets: free robots (those not yet occupying a pattern point) and free pattern points. For each pair (robot, point) it computes a tuple (distance, angle) and selects the lexicographically smallest tuple, i.e., the robot‑point pair with minimal distance and, among those, minimal angular deviation from the positive X‑axis. If the straight line segment does not intersect the inner safety circle C(O, d + ε), the robot moves directly; otherwise, the algorithm computes tangents from the robot and the target point to the safety circle, determines intersection points, and selects a detour that minimizes total travel while staying outside the safety region. Only one robot moves at a time, and the chosen detour guarantees that no other robot lies on the path (Lemma 7).

The overall execution proceeds through six high‑level steps:

1. If the origin O is empty, a robot at O moves a tiny distance ε along the positive X‑axis.
2. If O is a pattern point, the nearest robot moves onto O (ties broken by the global ordering).
3. Apply MoveRadiallyOut to push interior robots outward, creating a clear annular region.
4. Move the designated robot r₁ onto the nearest pattern point p₁.
5. Fill any remaining free pattern points on the SEC using interior robots first; if none remain, robots on the SEC shift to occupy those points without moving the SEC itself.
6. Finally, invoke MoveToDestination for all remaining robots, guaranteeing that each reaches its assigned pattern point.

Correctness is established through a series of lemmas. Lemma 1‑3 ensure a unique, globally shared coordinate system for the pattern. Lemma 4‑5 extend this uniqueness to the robot swarm. Lemma 6 proves that the radial outward movement never creates collisions and maintains the invariant that all robots stay within the annulus between the original SEC and the safety circle. Lemma 7 shows that the final assignment phase is collision‑free, respects the agreed coordinate system, and terminates in finite time because each iteration reduces the number of free robots by at least one.

Complexity analysis indicates that each robot needs to compute distances and angles to O and to all pattern points, leading to O(n²) computational effort overall, while the number of asynchronous rounds is bounded by O(n) because at most one robot moves per round in the critical phases.

The paper discusses limitations: the model assumes infinite visibility, point robots (no physical size), and a fully asymmetric initial configuration. Real‑world extensions would need to handle limited sensing range, robot dimensions, obstacles, and possible symmetric initial configurations that may render some patterns unattainable. The authors suggest future work on incorporating these practical constraints and testing the algorithm on physical robot platforms.

In summary, the work presents a robust deterministic algorithm that enables a swarm of asynchronous, oblivious robots—without any common direction or chirality—to form any given asymmetric pattern in finite time, while guaranteeing collision‑free motion and preserving a globally consistent coordinate system throughout the process. This advances the theoretical foundations of pattern formation in the most restrictive robot model known to date.