Collective Motion of Swarming Agents Evolving on a Sphere Manifold: A Fundamental Framework and Characterization

Collective motion of self-propelled agents has attracted much attention in vast disciplines. However, almost all investigations focus on such agents evolving in the Euclidean space, with rare concern of swarms on non-Euclidean manifolds. Here we present a novel and fundamental framework for agents evolving on a sphere manifold, with which a variety of concrete cooperative-rules of agents can be designed separately and integrated easily into the framework, which may perhaps pave a way for considering general spherical collective motion (SCM) of a swarm. As an example, one concrete cooperative-rule, i.e., the spherical direction-alignment (SDA), is provided, which corresponds to the usual and popular direction-alignment rule in the Euclidean space. The SCM of the agents with the SDA has many unique statistical properties and phase-transitions that are unexpected in the counterpart models evolving in the Euclidean space, which unveils that the topology of the sphere has an important impact on swarming emergence.

Scientific RepoRts | 5:13603 | DOi: 10.1038/srep13603 The main contributions of this paper are in the following two aspects. 1. In this paper, we are first interested in modeling a simplest possible yet effective framework for the SCM of multiple agents driven by a generic cooperative-rule (GCR), in which a set of the framework-rules describes spherical motion of each agent evolving on its own great-circle at each step, with a structure for integration of the GCR into the framework. As a result, the framework is versatile in the sense that: any instance of the GCR (or called a concrete cooperative-rule) can be then designed separately and integrated easily within the framework. Moreover, design of a concrete cooperative-rule for the SCM is rather similar (to a certain degree) as for the case of agents in the Euclidean space, which will add further convenience. New notions, phenomena, and characterizations of the SCM are then provided that are distinct from the cases of the ECM. 2. As an important example of the GCR, the spherical direction-alignment (SDA) is then provided, which corresponds to the Euclidean direction-alignment (EDA) 22 (that was first presented by Vicsek etc., and has been widely adopted in the past decades in many disciplines) of agents that is fundamental for the ECM. The SCM with the SDA has many unique characteristics that are unexpectedly distinct from the counterpart ECM model with the EDA, which unveil that the topology of the sphere has an important impact on swarming emergence. are the X-Y-Z coordinates of position p i (k) in the Cartesian coordinates, respectively. Notice that notation p i (k) also represents the position vector that starting from the center of the sphere to the position of agent i, according to the context. Denote T i (k) as the tangent plane to the sphere at position p i (k).
We use the notion "direction" to express the heading-orientation of an agent in the 3D Euclidean space, which corresponds to the heading angle that is valid in the 2D Euclidean space, and the 3D direction is a necessity to avoid possible symmetry-breaking control laws 28 . The direction of agent i at step k, denoted as ( ) ∈  d k i 3 , implies that: 1. it is a vector with a unitary magnitude: ( ) = d k 1 i ; and 2. it is constrained on the tangent plane T i (k), i.e., it is perpendicular to the position of this agent: The adaptive velocity of agent i has two meanings: 1. an adaptive direction according to a certain rule; and 2. an adaptive speed (here speed means the magnitude of velocity) v i (k), typically measured by the step-size at each step in a discrete formulation.
For spherical motion of an agent, there are two required configuration constraints: the positionconstraint Framework. The framework, as one of the main focuses in this paper, is described as the iterative equations as follows. First, for clarity and conciseness, define the vectorial function ×     f : 3 3 3 as: where the notation ⋅, ⋅ means the inner product in the 3D Euclidean space, which has the physical meaning that Consider the framework-rules of the SCM of agents in a discrete formulation, the initial positions p i (0), directions d i (0), and speeds v i (0) of all agents are given for the start of the iteration. Without loss of generality, assume v i (0) = 0 for all i and the time interval Δ t is a unit, i.e., Δ t = 1. The next position p i (k + 1), direction d i (k + 1), and speed v i (k + 1) of every agent i at step k + 1 that starting from position p i (k) with direction d i (k) and speed v i (k), are described by is called the mapped-Euclidean-step-size that is defined as tan 0 6 i i which ensures the spherical-step-size of agent i to be the speed v i (k), i.e., the great-circle-distance (GCD) between two steps p i (k) and ; v 0 is a constant (v 0 ≪ r) that represents the maximum-possible speed of all agents; the exponent α ≥ 0 characterizes the adaptivity of speed of agents.
Generic Cooperative-Rule and Notion of Spherical Approaching-Direction. In the framework, ζ ( + ) ∈  k 1 i 3 represents the GCR of agent i at step k + 1: where   f : it is the very direction of agent j when this agent just reaches at next position p j (k + 1), while before it switches to its new direction d j (k + 1) at that position (the superscript " − " in ( + ) − d k 1 j means "the left limit" at that position, hence the word "approaching" is used in this notion), the invariant quantity is where notation × means the cross product of two vectors. As a comparison, the notion "approaching-direction" is not needed in the Euclidean space, since any agent moves in a straight line to its next position without direction change during the traveling of any adjacent steps, thus ( + ) − d k 1 j just reduces to be the starting direction d j (k) at position p j (k) in the Euclidean space.
In design of ζ i (k + 1), generally ζ ( + ) ≤ k 1 1 i is required; and ζ i (k + 1) ┴ p i (k + 1) is preferred, but not necessarily required, owning to the perpendicular ramification by rule (4) using the function f.
In the SCM framework and the GCR, note that Interpretations. First, every agent i in the swarm with rules (3)(4) and (6) is constraint to move on the sphere along the moving direction d i (k) [that starts at position p i (k) at step k with the speed (or the spherical-step-size) v i (k)] from position p i (k) to position p i (k + 1); and then it will move along the moving direction d i (k + 1) [that starts at position p i (k + 1) at step k + 1 with the speed v i (k + 1)] to the next position, and so on. The formulation is robust in the sense that, any uncertain or noise in the system (although which may possibly influence the behaviors of the SCM of the agents) will not drive the agents away from the sphere.
The direction d i (k + 1) of agent i is determined by the cooperative-rule ζ i (k + 1) and then calculated with the unitary and perpendicular ramification by rule (4).
The physical meaning of ζ i (k + 1) can be viewed as a certain form of the local polarization surrounding agent i (according to a concrete form of function f g ) when all agents are just approaching to their next positions at step k + 1 (refer to the notion spherical approaching-direction). The vectorial information of determines v i (k + 1) by rule (5). The physical meaning of exponent α > 0 implies that, when ζ i (k + 1) shows a strong polarization, i.e., ζ ( + ) → k 1 1 i , agent i naturally moves faster; while ζ i (k + 1) shows no or a weak polarization, i.e., ζ ( + ) ≈ k 1 0 i , agent i naturally hesitates and is confused to move at such singularity situation, thus v i (k + 1) ≈ 0; this phenomenon is called the adaptive velocity mechanism (AVM) 27,28 .
It is easy to express a constant-speed-motion of agents, which is the usual assumption in literature (the advantage of the AVM vs the constant-speed-motion is illustrated in 27 ). That is, let i 0 for all i, k that replacing the rule (5), thus the equations (3) (4) describe the agents with constant speed v 0 on the sphere, with the rule (6) for different i, k reduces to be a constant: The framework is versatile in that, a variety of concrete cooperative-rules of ζ i (k + 1) can be designed separately and integrated easily into the framework.
Finally, a sphere manifold is a closed set, thus the periodic boundary condition (e.g., in 22 and many other papers) for agents in the Euclidean space is not needed when considering SCM. Certainly, one may apply the periodic boundary condition or other boundary conditions to a designated region of interest on the sphere and investigate the agents evolving on that region instead of the whole sphere, this is out of the scope of this paper.

Formulation of the SDA
As an important example of the GCR, consider Note that the rule (12) calculates the average of the spherical approaching-directions (8) For convenience, the rule (12), together with the unitary and perpendicular ramification (4), is called the SDA (the noise will be discussed in the following). Its physical meaning is that: every agent adopts the very direction at each step, which is derived by averaging the spherical approaching-directions of its neighboring agents at that step, then with the unitary and perpendicular ramification.
Note that the SDA rule (12) is equal to the expression: As a comparison, recall the fundamental EDA in the Euclidean space, with its physical meaning as that: every agent adopts the average direction of its neighboring agents at each step (that was first presented by Vicsek etc. in 22 , and widely adopted in the past decades in many disciplines).

SDA With Noise.
Noise is a common factor in CM of agents. Expression of noise in the SDA is also complex, compared with the EDA in the Euclidean space.
Consider the noise in the spherical approaching-directions in calculating the SDA in (12), i.e., a random swing of ( + ) − d k 1 j with an angle ϑ ∈  on the tangent plane T j (k + 1) for each agent j and step k (refer to Fig. 1), where ϑ represents a white noise with the strength randomly distributed in the interval [− η, η], η ≥ 0 is a constant that models the strength of the noise, note that the unit of η is in radians in this paper. Denote

Reduction from local sphere to 2D Euclidean space
The topologies of the sphere and the Euclidean space are completely different but have some relations. Consider a very limited local region of the sphere, which seems "flat" and thus approximates to the 2D  Euclidean space, then for the agents on a local sphere, some mathematical expressions are similar to the counterparts in the 2D Euclidean space. As an extreme case, when the radius of the sphere is infinite, the agents evolving on a very local sphere is similar as in the 2D Euclidean space, in this case the vectors in the SCM model reduce to be the corresponding vectors in 2D Euclidean space, then the tangent plane T j (k + 1) is always the 2D Euclidean plane, and ( + ) ≡ ( )  as in most literature, which, however, is less effective (or even improper) for the SCM, especially when the agents are distributed on a large scale of the sphere that is comparable with its radius. For example, this improper aspect can be seen from one instance of the swarming evolution (refer to Fig. 2, with the trajectories of the agents and characteristics illustrated in Figs 3 and 4, respectively), in which the snapshots (Fig. 2) already show the very order of the agents, while the usual order-parameter φ(k) (refer to Fig. 4) is still very low.
One suggested order-parameter for the SCM is the norm of the mean angular-momenta of agents, i.e.,  as the rotation-axis, denote the principal plane ( ) k as the plane that is perpendicular to ω(k) and passes through the center of the sphere, denote the principal great-circle as the great-circle of the sphere on ( ) k . For the scale of spherical distribution of agents, one measure is the mean GCD of agents to p(k), i.e.,

Evolution of SCM with SDA
It is just as expected that, when the distribution-density of the agents is large enough and the noise is weak, the agents will exhibit ordered motion as a whole; otherwise, the agents will break into some ordered fragments at the very beginning of the evolution. Interestingly, for the evolution of the SCM with the SDA, there are some phenomenons and properties that are distinct from the cases of the ECM.
For example, the fragments will have more opportunity later to collide and merge into a whole swarm, due to the topology of the sphere.
Also, the SCM with the SDA has a distinct shrinking effect, which will make the scale of the swarming agents continuously shrinking during the evolution, even with a large noise (refer to the next section), i.e., the swarm on the sphere has ordered motion with a stronger tolerance of noise. Figure 2 illustrate one instance of the swarming evolution. During the evolution, the scale of the swarming agents converges, as shown in c(k) and e(k), with increasing φ m (k) and φ ζ (k) in Fig. 4, which are appropriate order-parameters for the SCM in this case; however, the emergence cannot be effectively reflected by the usual order-parameter φ(k) that is valid for the ECM [note that φ(k) → 1 only when the agents converge to a very limited local scale on the sphere with ordered motion]. When r 0 is small, the agents will converge approximately to move on the sphere with the trajectories that are approximately parallel to the principal great-circle. As r 0 is large, oscillator trajectories of some agents may appear [e.g., refer to curves of e 1 (k), e 2 (k) in Fig. 4, the definition of e i (k) is provided in Eq. (33)], except the agents that are very near to plane ( ) k . Generally the agents will never evolve to a consensus velocity (or direction) even without noise (except the trivial case that all the agents always occupy same position at each step), due to the topology of the sphere, this is different from the case of the Euclidean space.

Statistical Properties
For the statistical properties, consider the SCM at a certain terminal step k. Note that the order parameters evolve faster (Fig. 4) and then remain relatively stable, while the scale of the swarm shrinks continuously and slowly. In this paper, k = 600 (if without special mention) is set instead of a still larger value, since the properties are relatively stable and the SCM is a computationally incentive simulation.
The state of a swarm at step k provides that slice of the evolution. For clarity, denote the values of φ m (k), φ ζ (k), φ(k), e(k), etc., at the terminal step, as φ m , φ ζ , φ, e, respectively. Each of the order-parameters φ m (k), φ ζ (k), φ(k) reflects one aspect of emergence; in many cases, all these order-parameters are required (one serves as a compensation to another) to characterize the emergence. Figures 5 and 6 illustrates some statistical properties of the SCM as a function of noise η and r 0 . For the curves of φ m with different noise, the values of φ m first increase as r 0 increases, and reach the peaks as r 0 is around r p ≈ 0.4, and then decrease after that. From the simulation results, the transition at the peaks implies the transition of motion patterns that from the pattern of ordered fragments for r 0 < r p to the pattern of cohesion (i.e., without fragments) for r 0 > r p . The increase of φ m is expected as r 0 increases for r 0 < r p . While for r 0 > r p , the agents move as a whole without fragments; in this case, as r 0 increases, the scale e of the swarm increases, thus φ m decreases (due to the topology of the sphere). φ has a similar but delayed transition. The scale e first decreases and then increases. In Figs 5 and 6, φ ζ monotonically decreases as r 0 increases; note that ζ i measures the local order of agent i, thus for a small enough r 0 , no matter how many fragments, φ ζ always has a large value; in other words, φ ζ is valid and effective when the agents have no (or less) fragments (r 0 > r p ), in this case, as r 0 increases, the scale e increases, the local order ζ i decreases (due to the topology of the sphere), thus φ ζ decreases.
The exponent α = 0 makes the values of φ m , φ, and e (after the transitions) less influenced by the noise than the case of α > 0.
The effects of r 0 and v 0 are completely opposite, with respect to the influence on the scale of the swarm. For a larger enough r 0 , the increase of r 0 tends to increase the scale of the swarm (Figs 5 and 6). While as v 0 increases, the shrinking effect (since all the agents move on the respective great-circles at each step, and no two great-circles are parallel, which tends to make the trajectories of the agents continuously converge, due to the topology of the sphere) of the swarm is strengthened, thus the scale e decreases, and the order parameters increase (Figs 7 and 8). As v 0 is larger enough (e.g., Figs 7 and 8 as v 0 = 0.08) that overweights the effect of r 0 , the scale e decreases monotonically and φ m increases monotonically.
The order parameters and the scale of the swarm are less influenced by the number n of agents (i.e., the different density of the swarming agents) in the swarm as illustrated in Figs 9 and 10.
As the noise increases, the suggested order parameters φ m (k) and φ ζ (k) for the SCM decrease, and e increases, as expected. However, the usual order parameter φ(k) has a larger value for a larger noise for r 0 < r p in Figs 5 and 6, this is another perspective to show that the usual order parameter φ(k) is not a good order parameter for the SCM (also refer to the example in the second paragraph of Section V).

Conclusion
This paper provides a fundamental yet simplest possible and effective framework for the SCM of agents driven by a GCR, which is versatile in the sense that, a variety of concrete cooperative rules of agents can be designed separately and integrated easily into the framework. This paper also designs the SDA, and investigates the unique phenomenons and properties that are specific to the SCM, which unveils an impact of the topology of the sphere on swarming emergence. There are some directions for future investigation. For example, i) the SCM with the SDA has very rich dynamics, with more characteristics that need to be further investigated; ii) other concrete cooperative rules of agents for different motion patterns on a sphere will be considered in a future paper; and iii) the framework has important implications in analyzing the SCM of many types of self-propelled agents on a sphere and even continuum-flows (e.g., by a coarse-grained approximation) for further investigation.