Planning and Control of Swarm Motions as

ASME 2012 5th Annual Dynamic Systems and Control Conference joint with the JSME 2012 11th Motion and Vibration Conference DSCC2012-MOVIC2012 October 17-19, 2012, Fort Lauderdale, Florida, USA

DSCC2012-MOVIC2012-8831

PLANNING AND CONTROL OF SWARM MOTIONS AS DEFORMABLE BODIES Hossein Rastgoftar Mechanical, Materials and Aerospace Engineering Department University of Central Florida Orlando, Florida 32816-2450 E-mail: [email protected]

Suhada Jayasuriya Mechanical, Materials and Aerospace Engineering Department University of Central Florida Orlando, FL 32816-2450 E-mail: [email protected] easily put together. Such swarms of autonomous agents may be deployed to accomplish numerous challenging tasks such as terrain mapping, surveillance, and monitoring of our oceans for oil spills. Having such large number of agents in a team makes the system very robust because a swarm could potentially accomplish an objective in spite of failure of some of the tiny agents within a team or the swarm. However, task planning and control of such large swarms with limited communication and computation capabilities is a hard problem that is receiving attention under the umbrella of swarm intelligence. There have been many research efforts to simulate coordinated group movements and behaviors in computer animations and robotics applications. There exist good techniques for modeling individual behavior within a group, such as Reynolds' boids based on each agent solely observing its local environment [1-4]. Of particular interest is how local decisions are to be tailored in the absence of complete knowledge of the entire swarm being available to each agent. The state of the art is that although in some very special cases interesting convergence results have been established for swarm robotics, several fundamental questions still remain unanswered [5]: What is a meaningful characterization that captures the essential features of a large collective/swarm? How can we plan for its behavior? How can we generate provably correct steering controls for each agent so that a desired group behavior is achieved? How much communication between agents in the collective is essential?

ABSTRACT In this paper a continuum approach for controlling the motion of a swarm of particles (“agents”) is presented. The control objective is to move the swarm from an initial reference configuration to a final configuration possibly of a different shape and size, avoid obstacles and inter-agent collisions while satisfying hard constraints on agent kinematics. The agents are considered to be inside a rectangle and it is assumed that the task is to move the swarm so that at the final time the agents are confined to a rectangle of possibly different size and orientation. It is shown that the agents can locally control their motions so that a collision free transfer respecting all agent constraints can be achieved with minimal inter-agent communication. At the nucleus of this approach is the “deformation of the group shape from a given reference configuration to a desired configuration”. The key idea is to find an appropriate homeomorphism between the initial and final configurations that respect all agent constraints. We show that a class of homogeneous transformations has very beneficial attributes. In particular, each particle or agent has a well-defined path that is based solely on its reference position. It necessarily means that an agent does not have to know the location of any other agent once the motion map is made available to an agent. We emphasize: (1) that minimum to no communication between agents is required for its implementation, and (2) it is independent of the number of agents, meaning that the approach is completely scalable. These two attributes are major advantages that are not present in most currently known path planners for swarms. Presented will be simulation results to illustrate the key ideas of the proposed approach.

There has been a plethora of research on cooperative control of multi-agents systems in the last two decades. Recent reviews of this extensive literature can be found in Bullo, Cortés, and Martinez [6], Murray [7], and Ren and Beard [8]. The approaches that have been taken to address cooperative control may be primarily classified as centralized or distributed depending on the communication topology. Some common approaches include: Leader-follower, Virtual structure, Behavioral based, Potential functions, and PDE formulations.

INTRODUCTION With advances in technology it is conceivable that teams of hundreds of tiny ground, air, or underwater robots can be

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In leader-follower methods agents in a group will follow the trajectory of a leader. The idea there is that the follower agents track a reference trajectory as prescribed by the leader. The main limitation of leader follower schemes is that agents may not be able to maintain such a desired reference trajectory due to violation of individual agent constraints on dynamics and actuators. [9-13].

pop up along the way requiring online real time change of trajectories. When such popups occur the continuum approach proposed only needs communication with two neighboring agents to guarantee no collisions while reaching a desired terminal configuration for the swarm. As motivation for the work described we consider a swarm task of moving a large number of agents initially occupying a square region of center A and side “a” to a final configuration where agents occupy a square of side “b” with center B. In executing the task the swarm must overcome the obstacles 𝑂1 and 𝑂2 (see Fig. 3) along the way in the sense that (i) there should not be collisions with the obstacles, (ii) agents should have a minimum pairwise distance of ‘c’ to avoid inter-agent collisions, (iii) agents must stay within pairwise maximum inter-agent distance of “d” at all times throughout the maneuver, and (iv) each agent actuator constraints must not be violated. This scenario encompasses many of the issues that are fundamental to swarm robotics and our results are aimed at providing a novel approach to answer these questions.

In Virtual structure control [14-17] an entire formation is constrained to move as a virtual rigid body. A variation of virtual rigid formation control where the formation is allowed to change its configuration as a function of time was studied by Maithripala et al [18]. In the latter approach a reference trajectory is computed at a given instant in real time by taking into consideration the dynamic constraints of each individual agent and the formation configuration constraints. Behavioral based formation control has been studied for motion control of multi robot systems [19]. One of three strategies: 1- Unit-center-referenced (each robot set its own position based on the average of the position of its neighbors), 2- Leader-referenced (each robot communicates with a leader to set its position), and 3- Neighbor-referenced (each robot maintains its relative distance and orientation with respect to its neighbors) [19] is used in the behavioral approach.

We start from the basic premise that tasks for large collectives evolving in complex environments should be "qualitatively" specified. More specifically, we assume that: (1) a swarm can be naturally described in terms of primitives such as shape, and size, while the exact position or trajectory of each agent is of secondary importance for the overall group behavior. (2) missions involving multi-agent swarms in complex environments are specified in terms of regions that must be reached or avoided. The accomplishment of such tasks does not require exact values for swarm features, but rather a need for guaranteeing their inclusion in given sets. Consequently, proposed is an approach where swarm migrations are treated as motions of deformable continua. The advantage of treating a swarm as a deformable body is that it allows the possibility for a swarm to change shape appropriately to facilitate maneuvering around obstacles. For example, it does not suffer from the difficulty/inability of the virtual structure method to maneuver through narrow areas. In addition the proposed method uses a minimum amount of communication. More specifically, agents will communicate with three virtual or actual leader agents or two neighboring agents depending on the nature of the task and uncertain environment. It is further assumed that each has limited actuation capability. Presented in this paper is a specific mapping among many possibilities yet to be researched which has the feature that the satisfaction of actuator limits is reduced to the proper selection of the final time. The paper focusses on keeping the agents of a team together in the sense of keeping them in a region such as a disc, ellipse, rectangle, etc. without any need to give specific location for an agent within such a region. So as far as each collective is concerned we are not concerned about pre-specifying individual trajectories for agents. What is important is that each agent in a team move as a collective within their region to accomplish

Potential function approaches are based on providing a dynamical system description for the motion of robots where the models include terms to (i) aid the self organization of robots, and (ii) to follow an approximate path decided according to continuous abstractions. The advantage of this approach is that with this preliminary information the motion of a robot can be computed locally using the distances to neighboring members of the swarm. Thus, the swarm can execute its motion even when contact with an external guiding leader or agent is lost. By imposing a condition that the robots are pairwise repelled from each other when they are too close, or are attracted when the separation gets too big it is possible to stabilize a swarm. Inter-robot collision avoidance can also be achieved by adding gyroscopic terms to the individual robot velocities corresponding to the abstraction, which will make them "steer away" from each other when they come too close. Modeling swarm behavior as parabolic or hyperbolic spatiotemporal PDEs have been studied in [20-22]. An analytical approach for the control of first order PDEs was presented in [23, 24]. A PDE based approach for the formation control of swarm motions was recently reported in [25]. The approach proposed here is based on continua and can be centralized or decentralized. Swarm motion planning based on this approach may be achieved by none to communication with two neighboring agents depending on the extent of uncertainty in accurately knowing the local state of an agent. For the case of an apriority specified initial and final configuration for a swarm individual agent motion can occur with no communication among agents as long as no obstacles

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a global swarm behavior for the collection of teams in a swarm. The paper is organized as follows. In section II given is the problem formulation and the approach where we first derive the necessary kinematic maps which is followed by a kinetic analysis leading to sufficient conditions for the motion determined by the kinematic map to satisfy the actuator force constraints of each agent. In section III the proposed approach is utilized on a special case that was studied in [5] which was earlier stated as the motivation for the subject of this paper. As will be discussed in section IV our approach exposes many possible global trajectories for the swarm to easily negotiate the obstacles and other constraints. Conclusions are given in section V.

𝑩𝒕

𝑩𝟎 𝑒̂3 ⃗⃗ 𝒊 𝑹

II. PROBLEM FORMULATION AND APPROACH Consider a swarm of agents at rest in an initial configuration 𝐵𝑜 at time 𝑡 = 𝑡𝑜 as shown in Fig. 1. We are interested in moving this swarm of vehicles from the initial configuration 𝐵𝑜 to a new configuration, termed the current configuration 𝐵𝑡 . The current configuration differs from the initial configuration and we say that the swarm has undergone a deformation from 𝐵𝑜 to𝐵𝑡 . Again we assume that each agent only knows its current location with respect to a global reference frame which may be inferred by data fusion, onboard sensing, inter-agent communications etc. It is further assumed that the current configuration may be prescribed apriori so that the desired goal configuration is achieved. For example, a typical scenario might be where a vehicle swarm is originally in a square configuration of side a at 𝑡 = 𝑡𝑜 , that may be required to deform into an elongated rectangular shape with a different orientation and location so that the swarm may travel through a narrow passage. As long as the motion map (path) is a homeomorphism between the reference and the current configuration [26], each agent is guaranteed to occupy a unique position in any of the configurations, implying no two agents can occupy the same place. One particular class of feasible deformations is homogeneous deformations. ⃗⃗ 𝒊 be the position vector, relative to a fixed point O, of Let 𝑹 a typical agent within𝐵𝑜 . Now suppose that the agent that ⃗⃗ 𝒊 at time 𝑡 = 𝑡𝑜 in the reference occupies a position 𝑹 configuration moves so that at a subsequent time it occupies a ⃗ 𝒊 at time t. Let us now denote the position 𝒓 ⃗ 𝒊 of new position 𝒓 ⃗⃗ the agent at time t with respect to its reference position 𝑹𝒊 at ⃗⃗ 𝒊 , 𝑡). We can ⃗𝒊 =𝒓 ⃗ 𝒊 (𝑹 time = 𝑡𝑜 , by an equation of the form 𝒓 think of this relationship as a way of specifying the locations of agents in a given reference configuration with respect to a current configuration. The idea here is that once we know the reference position of each agent and the mapping between the reference configuration 𝐵𝑜 and the current configuration 𝐵𝑡 , the current locations of the agents can be immediately determined. So the key idea then is to see if the map defining the resulting motion can be determined in a meaningful way.

𝑒̂1

⃗𝒊 𝒓

𝑂

𝑒̂2

FIGURE 1. INITIAL AND FINAL CONFIGURATION OF SWARM MOTION WHICH ARE GUIDED BY THREE CIMMANDERS

II.1- A Special Homeomorphism for the motion of a deforming triangle: Consider a multi agent system consisting of n agents where the objective is to move these agents in such a way that no inter-agent collision happens during the motion. To avoid collision of agents, it is necessary that each agent stay separated by a minimum distance from its nearest agent. We start by first embedding the entire swarm inside a covering rectangle by imagining the existence of three virtual or real (leader) commander agents defining three vertices of the rectangle. Let the initial and final configurations of the rectangles be as shown in fig. 2. Points A, B, and C represent commander agents that guide the swarm of agents inside the rectangle ABCD. The agents that are initially inside the rectangle ABCD are required to occupy the interior of the deformed rectangle 𝐴1 𝐵1 𝐶1 𝐷1 at the final time. Kinematic Map: Our first task is to find trajectories of the agents which are completely determined by the initial and final positions of the 3 vertex agents with the condition that no inter-agent collisions occur. Suppose that the motion consists of rigid body translation and a deformation of the following form: 𝑥𝑘 𝑖 (𝑡) = 𝑄𝑘𝑙 (𝑡)𝑋𝑙 𝑖 + 𝑑𝑘 (𝑡) (1) where k is a free index taking on values 1, 2, and 3 for a general 3D motion, l is a dummy index, 𝑥𝑘 𝑖 (𝑡) the ccordinate k of the agent i in 𝐵𝑡 (swarm configuration at time t), 𝑋𝑙 𝑖 the coordinate l of the agent i in 𝐵0 (reference configuration), 𝑑𝑘 the coordinate k of the rigid body displacement vector, and 𝑄𝑘𝑙 (𝑡) is the Jacobian matrix. In this homogenous transformation the Jacobian matrix Q is only a function of time and does not vary spatially.

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⃗⃗ 𝑖 (𝒕)‖ = ‖𝑚𝑖 (𝑸̈(𝒕)𝑹 ⃗⃗ 𝑖 + ⃗⃗𝑫̈(𝒕))‖ ‖𝑭 A

B

C

D

⃗⃗ ̈ (𝒕)‖) ≤ 𝑓 ⃗⃗ 𝑖 ‖ + ‖𝑫 ≤ 𝑚 (‖𝑸̈(𝒕)‖‖𝑹 𝑖

(5)

𝐴1

𝐵1

where ‖. ‖denotes the Euclidian norm, m the mass of the heaviest agent, and 𝑓𝑖 is the maximum force that can be applied by agent i. This in turn lead to constraint that ⃗⃗̈ (𝒕)‖ ≤ 𝑓𝑖 ⃗⃗ 𝑖 ‖ + ‖𝑫 ‖𝑸̈(𝒕)‖‖𝑹 (6) 𝑚

𝐶1

𝐷1

Non Collision of Agents: The condition that agents don’t get closer than a separation ⃗ 𝑖 (𝒕) distance d throughout the motion is derived next. Let 𝒓 ⃗ 𝑗 (𝒕) be the position of the ith and jth (for i=1,2,…, n; and 𝒓 j=1,2, …, n; i≠j) agents, respectively, at any time t. The separation between the two agents can be written as ⃗⃗ 𝑖 (𝒕) − 𝑹 ⃗⃗ 𝑗 (𝒕)) . ⃗ 𝑖 (𝒕) − 𝒓 ⃗ 𝑗 (𝒕) = 𝑸(𝒕) (𝑹 𝒓 (7) To avoid collisions we require ⃗⃗ 𝑖 (𝒕) − 𝑹 ⃗⃗ 𝑗 (𝒕))‖ ≥ 𝑑. ⃗ 𝑖 (𝒕) − 𝒓 ⃗ 𝑗 (𝒕)‖ = ‖𝑸(𝒕) (𝑹 ‖𝒓 (8) 2

FIGURE 2. INITIAL (ABCD) AND FINAL COFIGURATION 1(A 1 B1 C1 D1) OF THE MULTI AGENT SYSTEM; TRAJECTORIES OF COMMANDING AGENTS COLORED RED

Equation (1) can be rewritten with respect to the orthogonal 𝑒̂1 -𝑒̂2 -𝑒̂3 coordinate system (shown in Fig. 1) as follows: ⃗⃗ 𝑖 + ⃗⃗𝑫(𝒕) ⃗ 𝑖 (𝒕) = 𝑸(𝒕)𝑹 𝒓 (2) We note that the Jacobian matrix 𝑸(𝒕) is positive definite.

2

A sufficient condition for eqn. (8) to hold is that 𝜆𝑚𝑎𝑥 (𝑄) ≤ 1. This guarantees that any two agents separated by a distance ‘d’ will always stay separated by at least the same distance throughout the motion described by the kinematic map Q.

If three vertices A, B, and C of the reference rectangle ABCD are transformed to three vertices A1 , B1 , and C 1 of an arbitrary parallelogram 𝐴1 𝐵1 𝐶1 𝐷1 by the Jacobian matrix Q ⃗ , respectively, then the and rigid body displacement vector 𝐷 vertex D of the rectangle ABCD will map to the vertex D1 defining the unique parallelogram A1 B1 C1 D1 . Based on this fact the following observations can be made.:

III. A CASE STUDY Consider a swarm of particles which is constrained to move in a rectangular domain in the plane as shown in fig. 3. Each agent is assumed to be a disk with radius 0.01 𝑚.There are two solid obstacles 𝑂1 and 𝑂2 in this region; and a swarm of agents is initially distributed in a 2 m ×2 m square with its center located at the origin. It is desired that the centroid of this square be located at the point (9,-4) at the final time. Each agent is assumed to be a particle mass 0.2 kg.

Observation 1: ⃗ are given, and the agents are If the elements of Q and 𝐷 bounded by a rectangle at𝑡 = 𝑡0 , then the agents will remain bounded by a rectangle at any time𝑡 > 𝑡0 . Observation 2: ⃗ are not given, then each agent If the elements of Q and 𝐷 inside the embedding rectangle can find its trajectory by communication with the three commanding agents located at the three of vertices of the commanding rectangle at any time 𝑡 ≥ 𝑡0 .

Additionally, the force that can be exerted by each agent must take a value in the range [-2, 2]. Initial configuration and the locations of the agents within the swarm are as shown in fig. 4. The objective is to move this swarm free of inter-agent collisions from its initial location to the final location while avoiding the two obstacles. From the geometry of the terrain shown there are many possible paths that the swarm can take as long as we allow deformation of the swarm shape to negotiate various space restrictions along the way. One such path is where the swarm passes above the rectangular obstacle O1 avoid the triangle O2 and go directly to the final position. We will execute this particular plan to accomplish the required task.

Force Constraints: From equations (2), the acceleration of each agent i is ⃗𝒓̈ 𝑖 (𝒕) = 𝑸̈(𝒕)𝑹 ⃗⃗ 𝑖 + ⃗⃗𝑫̈(𝒕) (3) ⃗⃗ 𝑖 on the i-th If each agent has mass𝑚𝑖 , then the force 𝑭 agent required to execute the motion defined by the above kinematic map is ⃗⃗𝑭𝑖 (𝒕) = 𝑚𝑖 (𝑸̈(𝒕)𝑹 ⃗⃗ 𝑖 + ⃗⃗𝑫̈(𝒕)) (4) from which it follows that

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rectangular swarm shape at the entry into the passageway at the top of the rectangle obstacle. III. 1 Motion Map: The motion map is chosen as 𝑜1

[ 𝑜2

𝑥𝑖 (𝑡) 𝑄 (𝑡) 0 𝑋 𝐷 (𝑡) ] = {[ 11 ] [ 𝑖 ] + [ 11 ]} 0 𝑄22 (𝑡) 𝑌𝑖 𝑦𝑖 (𝑡) 𝐷21 (𝑡)

(9)

diagonal elements 𝑄11 (𝑡) and 𝑄22 (𝑡) corresponds to the deformation; 𝐷11 (𝑡) and 𝐷21 (𝑡) define the translation of the whole swarm To avoid excessive accelerations so that the force constraints may be satisfied throughout swarm motion, we assume the velocity and acceleration of the centroid of the moving rectangle to be zero at the initial configuration, the final configuration and at the intermediate points of the path at which the direction changes. One of the advantages of the method is that there are many possible functions of time which can satisfy these requirements and possibly others similar to them. For purposes of clarity and simplicity, it is assumed that every straight line segment of the swarm motion evolves as a cycloid in time. Different segments of the straight line maneuvers of the swarm are given by points P, Q and R. In addition, the velocity and acceleration of each agent is forced to be zero at the beginning and the final time of a deformation.

FIGURE 3. RECTANGULAR DOMAIN OF THE SWARM MOTION, AND INITIAL AND FINAL POSITION OF THE SWARM

Q

P

R FIGURE 4. INITIAL CONFIGURATION OF THE SWARM; INITIAL Remark: It should be pointed out that the same approach can be easily adopted to follow any other strategy to negotiate the terrain specified. We contrast this to the trajectory arrived at in [5] where the swarm heads down first and penetrate through the opening between the two obstacles to finally reach the desired location (see fig. 11). It should be pointed out that the latter trajectory can easily be accomplished with our method and is one of many feasible paths. We start by observing that the initial square region needs to first contract in both 𝑥 and 𝑦 directions to penetrate through the opening at the top of the rectangular obstacle. This is accomplished by defining the size and orientation of the

FIGURE 5. THE SWARM TRAJECTORY OF THE MID POINT OF THE EMBEDDING REACTANGLE

Deformation:

The motion strategy consists of first contracting the initial rectangle in the horizontal and vertical directions. For the contraction,𝑄11 (𝑡), 𝑄22 (𝑡), 𝐷11 (𝑡) and 𝐷21 (𝑡) can be chosen as

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𝑄11 (𝑡) = (1 −

0.5 2𝜋𝑡 2𝜋𝑡 ( − sin ( ))) 2𝜋 𝑇𝐷 𝑇𝐷

(10)

𝑄22 (𝑡) = (1 −

0.55 2𝜋𝑡 2𝜋𝑡 ( − sin ( ))) 2𝜋 𝑇𝐷 𝑇𝐷

(11)

𝐷11 (𝑡) = 0 𝐷21 (𝑡) = 0

𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 𝑇𝐷

translation (along the path shown in fig. 5), the swarm configuration does not change. We note that 𝑇𝐷 can be selected such that the force applied on each agent does not exceed the limit. Since

(12) (13)

FIGURE 7-B. CONFIGURATION OF THE SWARM AT THE SAMPLE 𝑡 = 0. .6 DURING DEFORMATION

FIGURE 6. THE CONFIGURATION OF THE SWARM INSIDE THE COMMANDING RECTANGLE AT FOUR TIME t=0 s, t=0.3 s, t=0.6 s and t=1s

FIGURE 7-C. CONFIGURATION OF THE SWARM AT THE SAMPLE 𝑡 = 1 DURING DEFORMATION

rectangle at four sample times𝑡 = 0 𝑠,𝑡 = 0.3 𝑠, 𝑡 = 0.6 𝑠 and 𝑡 = 1 𝑠 are shown together in in the fig. 5.

FIGURE 7-A. CONFIGURATION OF THE SWARM AT THE SAMPLE 𝑡 = 0.3 DURING DEFORMATION

the acceleration of the ith agent 𝑎𝑖 during the deformation step is less than

where 𝑇𝐷 is the time period of deformation. The configurations of the swarm, surrounding rectangle and once the swarm reaches the configuration shown in fig. 7-c, it follows the path shown in fig. 5. During this rigid body

2

2

𝑚𝑎𝑥 {√(𝑄̈11 (𝑡)) + (𝑄̈22 (𝑡)) },

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The configurations of the swarm at sample times𝑡 = 0.3 𝑠, 𝑡 = 0.6 𝑠 and 𝑡 = 1 𝑠 are also separately illustrated in figs: 6a, 6-b and 6-c.

[

𝑥𝑖 (𝑡) 𝐷 (𝑡) ] = [ 11 ]. 𝑦𝑖 (𝑡) 𝐷21 (𝑡)

(16)

And

FIGURE 9. THE MAGNITUDE OF THE REQUIRED FORCE OF THE SWARM DURING RIGID BODY TRANSLATION (-0.87,0.95)

(-0.87,0.95)

FIGURE 8. THE VARIATION OF THE EIGENVALUES OF THE JACOBIAN MATRIX. 2 2 2𝜋√0.52 + 0.0.552 2𝜋𝑡 √(𝑄̈11 (𝑡)) + (𝑄̈22 (𝑡)) = | | sin ( ) 𝑇𝐷 𝑇𝐷 2 4.6703 ≤ . (14) 𝑇𝐷 2 The maximum required forces (𝑓𝐷 ) applied to each agent is 4.6703 0.9341 𝑓𝐷 = 𝑚𝑖 = . (15) 𝑇𝐷 2 𝑇𝐷 2 Hence choosing 𝑇𝐷 = 1𝑠 leads to |𝑓𝐷 | < 2. In addition, the variation of both eigenvalues of the Jacobian matrix 𝑄 versus time is shown in fig. 8. The particles are initially distributed randomly but safe region 𝛺 must be considered around each agent at the initial configuration in order to avoid collision during the contraction. The transformation matrix is considered to be diagonal and contracting. Since the mapping is contracting the safe region 𝛺 is surrounded by a characteristic disc of radius ‘d’.

(0.14,-0.88)

(0.23,-0.34)

FIGURE 10. THE MAGNITUDE OF THE REQUIRED FORCE FOR CONTRACTION OF PARTICLES WHISH ARE INITIALLY LOCATED(0.14, −0.88), (0.23, −0.34) and (−0.87,0.95)

The rigid body translation of the center of the deformed rectangle comprises of the following stages:

It can be deduced from the transformation matrix Q that the max eigenvalues of the mappings are bounded as

1- Going on a straight line from point (0,0) to point (1.8,1) with the length of path

0.5 ≤ 𝜆𝑥 ≤ 1 0.45 ≤ 𝜆𝑦 ≤ 1

= √1.82 + 12 = 2.0591 𝑚 2- Going along the horizontal line from point (1.8,1) to point (7,1) with the length of path 2 = 7 − 1.8 = 5.2 𝑚 3- Going along the straight line from point (7,1) to point (9, −4) with the length of path 1

where 𝜆𝑥 and 𝜆𝑦 refer to the horizontal and vertical contraction respectively. Rigid Body Translation: The second step is to move the swarm above the obstacles until it reaches the final position. Since this is accomplished by rigid body translation the displacements of all agents are

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3

= √22 + 52 = 5.3852 𝑚

Therefore, the accelerations of the swarm and the necessary for each agent during the stages 1, 2 and 3 follows: We choose 2𝜋 𝑖 2𝜋𝑡 𝑎𝑖 = 2 sin2 ( ) 𝑇𝑖 𝑇𝑖 𝑓𝑜𝑟 𝑖 = 1, 2, 3 with the corresponding force magnitude 0.4𝜋 𝑖 2𝜋𝑡 2 𝑓𝑖 = 𝑚𝑖 𝑎𝑖 = ) 2 sin ( 𝑇𝑖 𝑇𝑖

forces are as (17)

(18) Z

𝑓𝑜𝑟 𝑖 = 1, 2, 3 where 𝑇𝑖 is the time period of the ith stage of the rigid body translation. Choosing 𝑇1 = 1.5 𝑠 and 𝑇2 = 𝑇3 = 2 𝑠 satisfy the force constraints |𝑓𝑖 | ≤ 2 𝑓𝑜𝑟 𝑖 = 1, 2, 3. 0.4𝜋 1 2.5875 𝑓1 𝑚𝑎𝑥 = = 𝑇1 2 𝑇1 2 0.4𝜋 2 6.5345 𝑓2 𝑚𝑎𝑥 = = 𝑇2 2 𝑇2 2 0.4𝜋 3 6.7672 𝑓3 𝑚𝑎𝑥 = = 𝑇3 2 𝑇3 2

FIGURE 11. in the DIFFERENT TRAJECTORY WAS PROPOSED IN [5]

To show that no collision happens during the contraction, the mapping of the circular region initially occupied by the agent initially located at (0.1054, −0.6041) is obtained at the sample times 0.3 𝑠, 0.6 𝑠 and 1 𝑠 and shown in figures 12-a, 12-b and 12-c respectively. Agents with their locations in the initial configuration at (0.14, −0.88), (0.23, −0.34) and (−0.87,0.95) are shown in fig. 9. The total period of the swarm motion from start to finish is 𝑇 = 𝑇𝐷 + 𝑇1 + 𝑇2 + 𝑇3 = 6.5 𝑠.

IV. SIMULATION AND DISCUSSION Several simulations are included to show the various aspects of the approach. First shown in fig. 5 are several positions of the swarm during rigid body translation from the starting square area to the final square area. Figures 7-a, 7-b and 7-c show the relative locations of the agents as they evolve during the contraction. The magnitude of the force required by those The agent initially located at (−0.87,0.95) requires more force than the agent initially located at (0.23, −0.34) for contraction. Also, the agent located at (0.14, −0.88) need the least force during contraction in comparison to the other two agents. We note that the trajectory taken by the swarm is given by the straight line paths of Fig. 5. Compared to this path the path given in [5] is what is shown in fig. 11. We emphasize that it is straightforward to construct a similar trajectory using the approach proposed in this paper. For example, we could move the swarm as a square from its initial position to a location directly under the opening between the two obstacles at Z. Then contract the swarm to fit through the opening and push it through the opening followed by a curvilinear translation to the final location.

FIGURE 12-A. MAPPING OF THE CIRCULAR REGION INITIALLY OCCUPIED BY THE AGENT INITIALLY LOCATED AT (0.23, −0.34) AT THE SAMPLE TIME0.3 𝑠

Figure 9 shows force variation over the rigid body translation segment of the swarm motion which occurs over the time interval 𝑡𝜖[1,6.5] 𝑠.

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Shown in fig. 9 are the required time varying forces over the segments 1 (the straight line between (0,0) and (1.8,1)), 2 (the straight line between (1.8,1) and (7,1)) and 3 (the straight line between (7,1) and (9, −4)). The forces that are required to effect the contraction depend on their respective initial and final configurations. The required force for the contraction map for agents which are initially located at(0.14, −0.88), (0.23, −0.34) and (−0.87,0.95) are shown in fig. 10.

Figures 12-a, 12-b and 12-c show that the region is initially occupied by the agent initially located at (0.1054, −0.6041) is always inside the moving disk with radius 𝑟 = 0.01 𝑚 which guarantees that no collision occurs during the contraction. V. CONCLUSION AND DISCUSSION An approach based on some basic ideas from continuum mechanics is given for the planning and control of swarm motions. The approach does not require inter-agent communication which makes the method very attractive for control and planning of swarms consisting of hundreds of agents. The swarm motion needs some communication to be received from a few command agents (real or virtual) but that can be kept to a minimum of 3 in many cases. There is only one piece of information that each agent must know and that is the motion map that takes them to the desired swarm configuration from its current configuration. The question remains however that the motion map either has to be made available to an entire team or each member of the team should be able to compute it locally. We are currently investigating such questions of communication as well as the question of how does a swarm adjust its mission when obstacles pop up along its way to a goal position. REFERENCES [1] Lewis M. A.; Bekey G. A., 1992, “The Behavioral SelfOrganization of Nanorobots Using Local Rules”, Proceedings of the 1992 IEEE/RSJ International Conference on Intelligent Robots and Systems.

FIGURE 12-B. MAPPING OF THE CIRCULAR REGION INITIALLY OCCUPIED BY THE AGENT INITIALLY LOCATED AT (0.23, −0.34) AT THE SAMPLE TIME0.6 𝑠

[2] Miller P., 2010, “The Smart Swarm: How Understanding Flocks, Schools, and Colonies Can Make Us Better at Communicating, Decision Making, and Getting Things Done.” New York: Avery. [3] Li W. and Shen W., 2011, “Swarm Behavior Control of Mobile Multi-Robots with Wireless Sensor Networks”, Journal of Network and Computer Applications, Volume 34, Issue 4, Pages 1398-1407. [4] Eren T., Belhumeur P. N., and Morse A. S., 2002,“Closing Ranks in Vehicle Formations Based Rigidity,” in Proc. IEEE Conf. Decision Control, Las Vegas, NV, vol. 3, pp. 2959–2964. [5] Kloetzer M. and Belta C., 2007, “Temporal Logic Planning and Control of Robotic Swarms by Hierarchical Abstractions”, IEEE Transactions on Robotics, vol. 23, No. 2. [6] Bullo F., Cortez J., and Martinez S., 2009, “Distributed Control of Robotic Network,” in Applied Mathematics Series, Princeton University Press. [7] Murray R. M., 2007, “Recent Research in Cooperative Control of Multivehicle System,” Journal of Dynamic Systems, Measurement and Control, 129, 571-583.

FIGURE 12-C. MAPPING OF THE CIRCULAR REGION INITIALLY OCCUPIED BY THE AGENT INITIALLY LOCATED AT (0.23, −0.34) AT THE SAMPLE TIME1 𝑠

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