Obstacle and Collision Avoidance Control Laws of a Swarm of Boids Bibhya Sharma, Jito Vanualailai, Jai Raj Abstract—This paper proposes a new obstacle and collision avoidance control laws for a three-dimensional swarm of boids. The swarm exhibit collective emergent behaviors whilst avoiding the obstacles in the workspace. While flocking, animals group up in order to do various tasks and even a greater chance of evading predators. A generalized algorithms for attraction to the centroid, inter-individual swarm avoidance and obstacle avoidance is designed in this paper. We present a set of new continuous time-invariant velocity control laws is presented which is formulated via the Lyapunov-based control scheme. The control laws proposed in this paper also ensures practical stability of the system. The effectiveness of the proposed control laws is demonstrated via computer simulations. Keywords—Lyapunov-based Control Scheme, Motion planning, Practical stability, Swarm. I. I NTRODUCTION A platform of biologically inspired concepts and behaviors has been applied into the real life situations. This has inspired researchers since numerous problems can be solved without rigorous mathematical approaches. It is basically the category of algorithms that imitate the way nature performs. This set of algorithms falls under various categories such as Artificial neural networks, Genetic algorithms, Evolutionary algorithms, Particle swarm optimization, Ant colony optimization, Fuzzy logic and others [1]. These have mostly prevailed in the field of robotics where solutions are sought for dull, dirty, difficult or dangerous tasks. The swarm behavior and its principles are now being used by scientists and researchers in many new approaches such as in optimization and in control of robots [2], [3]. The use of robots with the concept of swarming is significantly increasing in the manufacturing arena, not only for productivity enhancement but also for greater versatility and flexibility [4]. There is also the issue of limited resources which has brought about the use of multiple agents instead of single individuals. The advantages include flight control, satellite clustering, exploration, surveillance, foraging and cooperate manipulation [5], [6]. The applications of foraging could involve search-and-rescue teams at disaster sites. Teams of robots could be deployed to collect hazardous materials after a spill, nuclear reaction or other accidents in minimal time, hence, saving further loss in the environment. All in all, team(s) of homogeneous (even heterogeneous) robots working towards a common objective can satisfy stringent time, manpower and Bibhya Sharma is with the School of Computing, Information & sharma Jito Vanualailai and Jai Raj are also with the School of Computing, Information & Mathematical Sciences, University of the South Pacific. monetary demands, enhance performance and robustness, and harness desired multi-behaviors, each of which is extremely difficult if not entirely impossible to obtain from single agents [7]. The artificial potential field method has been frequently utilized to solve a wide range of problems permeating from robotic applications. Algorithms in this category tend to use physical analogies to establish artificial potential fields with repulsive fields around obstacles and attractive fields around goals [8]. A collision free path is determined by how much the robot is attracted to or repelled by the poles. The governing principle behind the artificial potential field method is to attach attractive field to the target and a repulsive field to each of the obstacles. Artificial potential fields methods have several advantages, the most important one being the easier implementation. The other advantages includes easier analytic representation of system singularities, limitations, and inequalities, its simplicity, favorable processing speeds, decentralization and scalability features that outweighs other methods [9]. This paper considers the motion planning and control of the swarm model when obstacles are introduced into the workspace. That is, we construct a Lyapunov-like function via the LbCS that guarantees the emergent behavior arising from the swarm, considering all practical limitations and constraints due to fixed obstacles. II. A THREE-DIMENSIONAL SWARM MODEL AND ITS PRACTICAL STABILITY At time t ≥ 0, let (x i (t),y i (t),z i (t)), i =1, 2,...,n, be the planar position of the ith individual, which we shall define as a point mass residing in a disk of radius r i > 0, b i =  (z 1 , z 2 , z 3 ) ∈ R 3 :(z 1 − x i ) 2 +(z 2 − y i ) 2 +(z 3 − z i ) 2 ≤ r 2 i  . Using the above notations, we have a system of first-order ODEs for the ith individual, assuming the initial condition at t = t 0 ≥ 0: x  i (t)= v i (t) y  i (t)= w i (t) z  i (t)= u i (t) x i0 := x i (t 0 ),y i0 := y i (t 0 ),z i0 := z i (t 0 ). ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (1) Mathematical Sciences, University of the South Pacific, Suva, Fiji (e-mail: b@usp.ac.fj). World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Natural and Physical Engineering Vol:8, No:2, 2014 253 International Scholarly and Scientific Research & Innovation 8(2) 2014 International Science Index Vol:8, No:2, 2014 waset.org/Publication/9997323