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Performance Metrics of Collective Coordinated Motion in Flocks Jorge L Performance Metrics of Collective Coordinated Motion in Flocks Jorge L. Zapotecatl1;2;3, Angelica´ Munoz-Mel˜ endez´ 2, and Carlos Gershenson3;4 1Posgrado en Ciencia e Ingenier´ıa de la Computacion,´ Universidad Nacional Autonoma´ de Mexico.´ 2Instituto Nacional de Astrof´ısica, Optica´ y Electronica´ 3Instituto de Investigaciones en Matematicas´ Aplicadas y en Sistemas, Universidad Nacional Autonoma´ de Mexico.´ 4 Centro de Ciencias de la Complejidad, UNAM, Mexico.´ email: [email protected], [email protected], [email protected] Downloaded from http://direct.mit.edu/isal/proceedings-pdf/alif2016/28/322/1904198/978-0-262-33936-0-ch054.pdf by guest on 25 September 2021 Abstract The measurement and evaluation of the collective perfor- mance of autonomous agents is an open issue (Navarro and Measurements of coordinated motion in flocks are necessary Mat´ıa, 2009). A lot of work remains to be done in the de- to evaluate their performance. In this work, a set of quantita- velopment of indicators that capture important aspects of the tive metrics to evaluate the performance of the spatial features exhibited by flocks are introduced and applied to the well- collective dynamics of groups of autonomous entities based, known boids of Reynolds. Our metrics are based on quanti- for instance, on the goal achievement, the formation of spa- tative indicators that have been used to evaluate fish schools. tial patterns, and the exploitation of resources. These indicators are revisited and extended as a set of three The tune of values for the achievement of coordinated new metrics that can be used to evaluate and design flocks. flocks is far from being trivial, because the parameters that Keywords. metrics, flock, behaviors, boids influence the behaviors of the members of flock are closely interrelated. Performance metrics are the criteria that deter- Introduction mine success in the behavior of a system. Therefore it is necessary to design objective performance metrics that al- An intriguing collective phenomenon of nature is caused low discern that flocks behave better. by animals moving as a coordinated unit, for example a In this work we propose new metrics in order to capture flock of birds or a fish school. This phenomenon has at- the global performance, in spatial terms, of the flocks such tracted interest in various scientific fields such as biology, that they can be used as a benchmark. Our metrics are based artificial life and artificial intelligence distributed, because in quantitative indicators that have been used to characterize the phenomenon suggests an intelligent organization that spatial features exhibited by a fish school: extension, polar- transcends the abilities of each individual (Camazine et al., ization and frequency of collisions (Huth and Wissel, 1992; 2003). Zheng et al., 2005). These measures are revisited and ex- The coordinated collective movement has important ap- tended in a set of three new measures: consistency in expan- plications such as virtual reality and computer animation. sion, consistency in polarization, and quality. The flocks models are often used to provide realistic-looking representations of flocks, schools or herds. For instance, Related Work they are used in the Half-Life video game to model the flying bird-like creatures or in the film Batman Returns to model A seminal work that aims at reproducing the flock phe- bat swarms and armies of penguins (Bajec and Heppner, nomenon is the model of coordinated collective motion pro- 2009). posed by Reynolds (Reynolds, 1987), where individual enti- In addition, the flocks models can be used for direct con- ties, generically known as boids, achieve realistic behaviors trol and stabilization of teams of simple Unmanned Ground by the application of a set of basic rules. This model is used Vehicles (UGV) or Micro Aerial Vehicles (MAV) in swarm in this work to applied our performance metrics. Therefore, robotics (Min and Wang, 2011; Saska et al., 2014). Also, we review in detail this model in the next section. they can be used in applications that require the coordinated In the area of biology, Huth and Wissel (Huth and Wissel, action of multiple autonomous individuals such as flying 1992) present a simulation of a school of fish. The behav- robots (drones) used in collective search, agricultural moni- ior of each fish are attraction, repulsion, parallel orientation toring and event surveillance (Vsrhelyi et al., 2014). and search. Polarization and extension are proposed as de- Other interesting applications where flock models has scriptive metrics of a school. The polarization reflects the been applied is the automatic programming of Internet degree of alignment of the agents headings, if fish belong- multi-channel radio stations and for optimization tasks ing to the school are oriented in similar directions a school (Iba´nez˜ et al., 2003; Cui and Shi, 2009). has a small polarization. The extension reflects the degree of cohesion that has a school, that is, how far the fishes are vectors “forward”(x axis), “side”(z axis) and “up”(y axis). between themselves. Where each boid has a local view of its environment called In the work of Huepe and Aldana (Huepe and Aldana, “area of perception”related to a steering behavior. The area 2008) are compared three simple models that reproduce of perception is determined by a radius r and a angle θ (field qualitatively the emergent swarming behavior of bird flocks of view) where only neighbors who are in the area of per- or fish schools, by using of the metrics: polarization, local ception are selected for calculating certain steering behavior density and nearest neighbor mean distance. While the po- (see Figure 1). larization (standard order parameters use in the flocks) be- have equivalently in all cases, the local density and near- est neighbor mean distance introduced provide a more de- tailed description that can clearly distinguish the properties of swarming behavior. In the work of Navarro and Mat´ıa (Navarro and Mat´ıa, Downloaded from http://direct.mit.edu/isal/proceedings-pdf/alif2016/28/322/1904198/978-0-262-33936-0-ch054.pdf by guest on 25 September 2021 2009) a set of metrics that measures the performance of collective movement of mobile robots is proposed and dis- cussed. The different metrics proposed cover several aspects of the characteristics of the collective movement including: those related to the area and shape of the group; the move- Figure 1: Figure (a) shows the local space of a boid. Figure (b) ment; and the positioning and orientation of its members. shows the area of perception of a boid. In the work of Bajec et al. (Bajec et al., 2007) it is pre- sented an artificial animal construction framework that has The set of boids bj perceived by the boid bi is denoted Pi been obtained as a generalization of the existing bird flock- and is calculated as follows: ing models. A set of metrics that can measure and judge boids the flocking behavior of a group of is presented and Step 1. Calculate distance, distance dij is determined from the metrics are used in a series of controlled experiments to the boid bi and boid bj, evaluate the flocking behavior of boids. dij = k ~pj(t) − ~pi(t)k (2) The boids model Step 2. Calculate angle, the angle θij is determined between This work is based on a model of coordinated collective mo- the vector “forward”of boid bi and the unit vector in the di- tion proposed by Reynolds in his classic article (Reynolds, rection of the position of the boid bj, 1987), which is inspired by the movement of flocks and schools. Entities belonging to a formation (birds, fish, etc.) " T # were called by Reynolds generically as boids. Each boid 180 ~ ~pj(t) − ~pi(t) θij = arccos fi(t) · (3) apply a simple set of steering behaviors such as cohesion, π k ~pj(t) − ~pi(t)k separation and alignment, that govern their movement. A flock is the result of the interaction of each boid with its Step 3. Calculate neighbors, determine if the distance dij neighbors. is less than the radius r of boid bi and angle θij is within the angle θ of boid b , then the boid b belongs to its local The set B of n boids bi involved in the flock is denoted i j by formula 1. neighborhood, B = fbi; i = 1; 2; : : : ; ng (1) Pi = fbj 2 B; 8bj : dij < r ^ θij < θ; j = 1; 2; : : : ; mg (4) Each boid bi has a position vector ~pi(t) and a velocity where m is the number of boids perceived by the boid bi on vector ~vi(t) that describe its motion in space in a time t. its radius of steering behavior. The force which adjusts the speed of the boid is typically an impulse generated by the same, so that impulse is limited The steering behaviors of a boid bi executed at time t in a scalar of maximum force denoted as fm. In addition, are cohesion, ~ci(t); separation, ~si(t); and alignment, ~ai(t) boids are restricted to a maximum speed expressed as sm. where the areas of perception associated with these steering This speed limit is imposed by a gating of the velocity vector behaviors are determined by the radii and angles rc, θc; rs, of the boid. In addition, the acceleration acquires a boid θs; and ra, θa respectively. In addition, the steering behav- also depends on the inertia of the body expressed as a scalar iors cohesion, separation and alignment are associated with quantity mass mb. the sets of boids bj perceived Ci, Si and Ai, respectively. The local space associated with each boid bi is described The calculation of neighbors boids belonging to said sets is ~ by fi(t), ~si(t) and ~ui(t) which refer respectively to the performed as in steps 1, 2 and 3 above.
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