Technical Notes and Correspondence - IEEE Xplore

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 5, MAY 2014

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Technical Notes and Correspondence Unified Cooperative Control of Multiple Agents on a Sphere for Different Spherical Patterns Wei Li, Member, IEEE, and Mark W. Spong, Fellow, IEEE

Abstract—In this technical note, we consider the cooperative control of multiple agents on a sphere, provide the appropriate attraction-repulsion interactions between agents with introduction of the state-dependent repulsive coefficients, and design the unified control laws for agents to achieve three fundamental spherical configurations or patterns: rendezvous, uniform deployment, and formation, using both the first- and second-order models. We present the analysis of the stability, scaling and geometric patterns of the models on the sphere under different attraction-repulsion parameters, with illustration of the geometrical patterns and the geometrical isomerism by simulations. The closed-loop dynamics of agents are invariant with respect to group rotation actions on the sphere. Index Terms—Cooperative control, deployment, formation, manifold, motion patterns, rendezvous, spherical cooperation, spherical pattern.

I. INTRODUCTION The collective motion and cooperative control of multiple agents have attracted much interest, for example, flocking [1]–[3], consensus [4], synchronization [5], [6], rendezvous [7], [8], formation [9]–[11], deployment, coverage or partition [12]–[20], etc. Most work deals with the collective motion in the free Euclidean space. There has been relatively less research dealing with the agents moving in the Euclidean space with constraints [26] or on a non-Euclidean manifold, such as Lie groups [21], including SE(2), SE(3) [22]–[25] and the [27]–[29]. Moreover, most previous papers in this area circle have used first-order integrator models of the individual agents, one reason being that the second-order dynamics of the agents may not have much physical meaning in general on a non-Euclidean manifold. However, for the agents moving on the manifolds that can be viewed as a Euclidean space with some geometric constraints, then the second-order dynamics of the agents has more important implications when considering Newton’s second law, especially for the agents with inertia that can not be neglected. With regard to spherical motions of the agents, the control of swarms on a sphere is investigated in [26] with emphasis on synchronous behavior of first-order Kuramoto-like oscillators coupled on social networks. Quantized control, investigated in [20], can be used in the spherical multicentric problem or spherical partitions. The most recent series of investigations are reported in [31]–[33], where the agents travel at the same constant speed for circular motion on a sphere, which can Manuscript received August 02, 2011; revised November 04, 2012 and August 04, 2013; accepted October 15, 2013. Date of publication October 23, 2013; date of current version April 18, 2014. This work was supported by NSF under Grant 61104043. Recommended by Associate Editor M. Prandini. W. Li is with the Department of Control and Systems Engineering at Nanjing University, Nanjing, China (e-mail: [email protected]). M. W. Spong is with Eric Jonsson School of Engineering and Computer Science, University of Texas at Dallas, Richardson, TX 75080 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2013.2286897

be stationary [31] or rotating [32] with or without [31] the flow-field [32], [33] on its surface. The dynamics of the agents are under the natural Frenet-Serret frame [22]–[24], with the steering controls derived using the angles of the agents under the spherical coordinates, which is symmetry-breaking with non-uniform property (compare the lengths of the arcs with same latitude-longitude intervals in the polar region and the region near the equator of the sphere) and the singularities (polars). Thus the control laws [31]–[33] are symmetry-breaking and the behaviors of the agents are not invariant with respect to the group rotation action on the sphere. In the spherical motion of agents with first-order dynamics [26], the control laws are symmetry-preserving. However, the repulsion between agents decreases to zero when two agents approach to each other. Such repulsion borrowed from the repulsive oscillators [30] may be relatively easier to analyze, but may not be a good repulsion candidate; moreover, the agents with oscillator-like repulsions have difficulty to achieve uniform deployment on a sphere, even with all-to-all coupling. Also the interaction between any pair of agents is either attraction or repulsion [26], which makes it difficult to control the scaling of the agents on a local region of the sphere. In this technical note, we consider a class of the cooperative control laws for multiple agents evolving on a sphere, with the final configurations or patterns depending on the values of the control law parameters. The contributions in this technical note are mainly in three aspects: 1) we extend the work [26] by modifying the interactions between agents in the attraction-repulsion paradigm, particularly the introduction of the state-dependent repulsive coefficients that suitable for spherical evolution, for both the first- and second-order dynamics of agents, as compared with the first-order dynamics with constant repulsive coefficient in previous work; 2) we investigate the dynamics of the agents with symmetry-preserving controls on a sphere, as compared with the symmetry-breaking controls in [31]–[33]; and 3) we design the unified control law for different spherical patterns: i) rendezvous, i.e., convergence of agents to a common position, ii) uniform deployment, i.e., dispersal or distribution uniformly, and iii) formation, i.e., convergence of agents to certain shapes in a local region on the sphere. We provide the analysis of the stability, scaling, and geometric spherical patterns of the agents under different attraction-repulsion parameters. This technical note is arranged as follows: Section II describes the cooperative control of the agents on a sphere with the first- and secondorder models. Section III designs the control laws. Sections IV and V present the main results on the stability and scaling. Section VI is the characterization of the patterns on sphere. Section VII illustrates the geometrical patterns. Section VIII is the connection with the oscillators. Section IX is the conclusion. II. PROBLEM DESCRIPTION Consider homogeneous agents moving collectively in -dimen. We are interested in designing the sional Euclidean space, cooperative control laws so that the agents are constrained to move on with the radius for different spherical conthe sphere when figurations or patterns. The sphere reduces to be the circle , and the sphere is the usual sphere in 3D space. Without loss of generality, we assume that the origin of the inertial frame lies at the center of the sphere. Define the position of agent as (for clarity and simplicity, the parameter is omitted in all variables

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in the following, for example, is abbreviated as ), we require for all agents on the sphere. For the first-order dynamics, the desired state space of the agents is is required, the sphere; for the second-order dynamics, denotes the usual inner product in -D space, where the desired state space of the agents is the tangent bundle of the sphere . In the design of the control laws, the constraints for the agents confining on the sphere need to be considered. Definition 1: 1) The agents are said to Rendezvous if

Fig. 1. Illustration of

, the dotted circle is the great circle of the sphere.

(1)

where is the damping gain, and the control law satisfies the condition (6). Then, if agent initially lies on the sphere with , it is constrained to the sphere for all , i.e.,

in which case the final configuration is a Rendezvous Configuration. The rendezvous configuration is equivalent to where the Average Position of the agents is defined as:

(9)

(2) 2) The agents are said to Deploy if (3)

, then . From Proof: Define the definition of , we have Therefore, , thus the initial condition of the ODE implies , the result follows. Remark 1: The first term of in (8) can be interpreted as a centrifugal force acting on agent , the second term is a linear damping, and is the overall force on agent by other agents.

in which case the final configuration is called a Deployment. 3) The agents are said to Deploy Locally if

III. DESIGN OF CONTROL LAWS OF BOTH MODELS First define the vector

(4) , in which case the final configuration is called where a Formation or Local Formation. The above deployment definition is the weakest possible and does not say anything about the final configuration or pattern of the agents. We will give the results below that quantify the coverage of the deployment on the sphere as functions of the controller gains. Of course, the concept of the deployment on a sphere is more meaningful for a large number of the agents, as compared with only two agents that are dis). tributed at antipodal points (i.e.,

Consider the agents governed by the first-order integrators: (5)

when

is the control input of agent , and the initial condition satisfy . Agent is then constrained to lie on the sphere for , is always perpendicular to , i.e., (6)

B. Second-Order Model In reality, the physical agents are governed by second-order Newtonian (or Lagrangian) dynamics. For this reason, we also consider the second-order model: (7) where the structure of the control law Proposition 1: Let the control law

as illustrated in Fig. 1, we have or

.

, and its magnitude Definition 2: Define the instantaneous force agent as:

when . on agent due to

(11) is the attraction parameter (also attraction coefficient), is the repulsion parameter, is the state-dependent repulsive coefficient for the pair of agents , and is the repulsive term on agent due to agent . Remark 2: Note the usage of the terms “parameter” and “coefficient” in Definition 2. when and when , The force which is more suitable as the repulsion candidate (as compared with the , which approaches to zero as ). oscillator-like repulsion , found by setting the right hand The equilibrium positions of agents side of (11) equal to zero, have relative Euclidean distance . The term is a the function of the agents’ distance and the choice of , and is attractive, repulsive, or mixed, i.e., , , then is attractive, or ; 1) when , , then is repulsive and stronger for the 2) when ; refer to Fig. 2; agents closer together, , , it depends on the ratio : 3) when , is always repulsive, ; • if , then , and is repulsive when , • if and else attractive, refer to Fig. 3. in both In the remainder of this technical note, the control law i.e., models (5) (7) above is given as: where

A. First-Order Model

where

(10)

is given in Proposition 1. in (7) be given as

(8)

(12)

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Fig. 4. Illustration of

from position

• on the whole sphere Fig. 2. Illustration of (the blue arrows) and as the function of when agent at different locations, with assumption of the fixed position and . In the upper sphere, , . In the lower sphere, .

of agent to the average vector .

if

,

in this case ; • locally if , in this case . , , the agents rendezvous, . 3) For Proof: Consider the dynamics of the average position :

(14) Since (9) holds, then , we have the following three equations: of

Fig. 3.

as the function of

,

,

, and from definition

.

(15) where

is the neighbor set of agent , when , else . . Then is the overall attraction-repulsion force on , so the agents will remain on the sphere. agent , and and (i.e., without the stateWe note that, when ) coincides with the dependent repulsion), the control law (12) ( control law in Olfati-Saber’s work [26] for the first-order integrators:

(16) (17) then (14) is (18)

which allows mixed signs of , some positive and some negative, represents attraction, represents repulsion. where From (18), when Define

IV. ANALYSIS ON FIRST-ORDER MODEL Define as the Euclidean distance from position of agent to is a measure of the the average vector (Fig. 4). Then when . In the remainder of the scale of the pattern, and for all , i.e., the all-to-all coupling. technical note, we take Theorem 1: Consider the dynamics (5) with control input (12). , , the agents deploy on the whole sphere 1) For , and further, the average position of agents satisfies:

Define

as the angle between and (when ), , , then . Define where . when , . since , , cannot all be decreases as increases. Then perpendicular to ;

(13) with the constant convergence rate. , , then the agents deploy: 2) For

, (13) holds. is the only equilibrium. , from (18), we have

(19)

and

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1) When

, then

2) When and tial values

(note that

, then since . and (also

3) For , , then: when (note that the system), we have

) and

then

.

(note that ), in this case, or depends on the iniand ). Then we have . when , and is an unstable equilibrium of

, ,

, and we have ,

,

, so (22) .

Then based on (15), (16) and (22), , define the function 2) When

.

Remark 3: The agents cannot rendezvous when . In the deployment of agents in Theorem 1, generally, the agents have uniform deployment on sphere (Sections VI, VII). As a comparison: Proposition 2: Consider the first-order dynamics (5) of the agents with the control laws of only constant repulsive coefficient , i.e.,

where

, , , so , . In this case, the dynamics of the average

. Then , position

is:

(20) then

. Proof:

.

.

From (17) (19), , where since , , is an unstable equilibrium. . Remark 4: In Proposition 2, although , the agents can rarely even with all-to-all coupling, make the uniform deployment on (i.e., the constantthe same is for the reduction of the system on repulsive Kuramoto-like oscillators, refer to Section VIII).

And since

, , i.e.,

, then

then similar as Theorem 1, we have: 1) when , , so

V. ANALYSIS ON SECOND-ORDER MODEL

then

Theorem 2: Consider the dynamics (7) with given by (8) and (12). , , then the agents deploy on the whole sphere 1) For , . with , , then the agents deploy 2) For if ; • on the whole sphere , in this case . • locally if Proof: Since the agents are on the sphere, then for any agents , and

,

, so

; 2) when .

, then , ,

Theorem 3: Consider the dynamics (7) with the control input given , . Then the agents rendezvous, i.e., by (8) and (12). . , then Proof: Define . If there exists at least one such that does not hold for , then , so , . Otherwise , , . In , , that is, this case (23)

Define

then

1) When

, define

, we have

, define the function (21)

so parallels to , and (since is an unstable , and equilibrium of the system), then is not stable, so for all , and the result holds. Remark 5: The larger the value of , the more potential energy the agents have; the larger the value of , the faster the energy decays. So the agents rendezvous faster with less oscillations if is large enough, as illustrated in Fig. 5. Remark 6: In addition, the final rendezvous position of the agents and the initial conditions, but is not depends on the parameters , easily predicted, especially when the initial average position which needs further investigation. One simple case is that, if all the agents are initially located on a hemisphere of the sphere with zero velocities, then the final rendezvous position is on this hemisphere. Proposition 3: Consider the second-order dynamics (7) (8). Assume . the control input is given as (20), then , then Proof: Define . If , then in (23), which is . unstable. So Remark 7: The remark of Proposition 2 holds here.

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Fig. 5. Rendezvous on the sphere.

,

,

,

.

Fig. 6. Illustration of in (i)(ii) from two different view angles, the eight agents are on the vertices of two parallel squares with blue edges, and every ), every blue edge has the same length 1.1672 (this is the smallest distance in is illustrated in (iii), every blue dotted red edge has the same length 1.2939. edge has the same length 1.0515.

force

VI. CHARACTERIZATION OF PATTERNS ON SPHERE Proposition 4: The agents on the sphere always satisfy the equation:

as a result, the zero average position

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equals to the condition:

and then does not exactly hold (as a comparison, and in the case of the rendezvous), as the errors accumulate, the agents will finally escape from the sphere. Additionally, the disturbances in the system can also cause the agents to escape from the sphere so that additional feedback term is needed for keeping the agents on the sphere, which makes the control more robust. Thus for applications, the control law (12) becomes

where the feedback term for the agent confining on the sphere is: Proof: the sphere. And note that

for all agents

on . Then

. Definition 3: For the dynamics of the two models (5) and (7), define the distance-matrix:

which is a function of the numbering sequence of agents; while we that is same as the definition of matrix except use the pattern that the numbering sequence of agents is not considered. ) of the models, the matrix For given parameters ( may be different for different , and even the pattern may not be unique. Thus, we have the following notions: Definition 4: For any initial conditions of the agents, if the distanceis unique for a fixed numbering sequence of agents, matrix is said to be unique; else, is said to have the then the pattern : geometrical isomerism. For the unique pattern , then it is said the abso• if every agent has the equal status in lute-symmetric-pattern; ) have the equal status in • if only a part of agents ( , then it is said the partial-symmetric-pattern; else, , then it is said • if no any two agents have the equal status in the non-symmetric pattern. Proposition 5: One necessary condition for absolute-symmetric. patterns of the agents is that, the agents’ average position The uniform deployment means that the agents deploy evenly spaced on the sphere. Certainly any absolute-symmetric-pattern is the uniform deployment. However, for non-absolute-symmetric-patterns, the precise mathematical description of the characterization of the uniform deployment is difficult (see the examples in Section VII). VII. ILLUSTRATION OF PATTERNS ON SPHERE Mathematically, the agents with the control laws are always confined on the sphere, since for all . However, numerically the result ), in this case, does not hold when the agents have repulsions ( , the limited accuracy of computation leads to non-exact zero

(24) and theoretically, for the agent on the sphere, . Examples 1–4 illustrate the uniform deployment of the agents in The, , , , ; in this case, orem 2, ; the simulation runs 200 seconds for every example. is that ( ): 1) Example 1: One instance of matrix

every agent has four equally nearest neighboring agents. One instance is that: four agents at vertices of a regular square in of the pattern - plane, and two agents at , respectively. (refer to Fig. 6) is not 2) Example 2: It may be unexpected that is more a cube (i.e., not the absolute-symmetric-pattern). Actually, stable than a cube pattern with the sphere as its circumscribed sphere (every edge of the cube has length , which is smaller than in ). , where 12 Remark 8: A similar phenomenon can be found in agents on vertices of two hexagons do not form a polygonal column. , , 4, 6, 12, are the absolute-symThe patterns metric-patterns. , only for , the distances between Remark 9: When any two agents are same, i.e., for any numbering . For example, in , for . sequences is the absolute-symmetric-pattern (refer to 3) Example 3: Fig. 6). Every agent has five equally nearest neighbors, and the distance between any pair of the agents is either 2 or 1.7013 or 1.0515. (Fig. 7), the two blue hexagons are regular 4) Example 4: In and parallel, every blue edge has the same length; every red solid edge has the same length; and every red dotted edge has the same length. The two agents not on the hexagons are the antipodal points on the sphere,

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VIII. CONNECTIONS WITH OSCILLATORS ON CIRCLE For the dynamics (5) with the control input (12) in 2D space with , , i.e., when the additional assumption that underlying state is constrained directly on , then the agents become the oscillators, and the state of agent can be described by only its angle , it is positive in counter-clockwise and negative in clockwise. becomes , and the distance In this case, the vector becomes , where is the imaginary unit, the notation represents the magnitude of a complex value, note , then the square of the distance: that (25) Then corresponding to the dynamics of the agents (5) and (12), we have the reduction of Theorem 1: Proposition 6: Consider the state-dependent coupling oscillators: (26) Fig. 7. Illustration of ferent view angles. (i)(ii)

and

, each pattern is illustrated from two dif. (iii)(iv) .

. The oscillators will have rendezvous, deployment or as in Theorem 1. local formation with different values of , then (26) reduces to be Kuramoto-like oscillators [30]: Let (27)

Fig. 8. The trajectories and final pattern.

,

,

,

.

and has been widely investigated, e.g., in [27] ( ) with ( ) represents attraction (repulsion); in [26] with mixed signs of ( ): or represents attraction/repulsion. Compared with (26) (27), the repulsion in (27) may not be a good repulsion candidate: the closer the agents are to one another, however, ) rarely have the smaller repulsions between them; the agents ( uniform deployment even with the all-to-all repulsions. Similarly, the second-order dynamics (7) of agents confined on with the control law (8) and (12) can be described as the second-order dynamics of the oscillators with the state-dependent couplings: (28)

Fig. 9. Illustration of two formation patterns of agents with same initial condi, , . (i) . (ii) . tions, with

, which has similar physical meaning as of the dynamics (7) of agents with inertia in Theorem 2, and thus also deserves more attention. IX. CONCLUSION

i.e., their distance is 2. In , all the blue edges have same length, and all dotted red edges have same length; there are five squares (with that connected in a roll, refer to Fig. 7. solid edges) in Examples 5–8 illustrate the local formations in Theorem 2. has two different patterns or geometrical isomers 5) Example 5: from simulations: i) the pentagon pattern with all agents at its vertices, and ii) the pattern that four agents at the vertices of a square and one agent above its center of the square on the sphere. , , 7, 8, 9, is generally a pentagon, hexagon, 6) Example 6: agents at its vertices and heptagon or octagon respectively, with one agent above its center of the corresponding polygon. (Note that only the agents on the vertices of the polygon are in the same plane.) From Theorem 2, , for agents at

In this technical note, we consider the cooperative control of multiple agents on a sphere with the unified control laws for different configurations or patterns. The spherical cooperation has important implications and deserves more attention. Many questions remain unsolved, e.g., can the rendezvous position or the pattern’s center be predicted from the parameters and the initial condition? what are the symmetry-preserving controls for the circular, vortex and predefined patterns on a sphere? and is there any way to predict the final pattern of the uniform deployment of a large group of agents? The agents on a sphere with general coupling topology is also an important direction for future investigation.

. the vertices, and with 17 agents forming two con7) Example 7: Fig. 8 shows centric circles (or polygons) and one above their center. 8) Example 8: The relative positions of agents are determined by , , and the initial condition. Fig. 9 is an example.

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 5, MAY 2014

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