MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Ant Colony Optimization Ant Colony Optimization Algorithm and Approaches in Robot Path Planning Katinka B¨ ohm Universit¨ at Hamburg Fakult¨ at f¨ ur Mathematik, Informatik und Naturwissenschaften Fachbereich Informatik Technische Aspekte Multimodaler Systeme January 4th, 2016 Katinka B¨ ohm 1
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Ant Colony Optimization Structure 1. Introduction 2. Theoretical Approach 3. Robot Path Planning with ACO 4. Analysis 5. Interesting Applications 6. Recap 7. References Katinka B¨ ohm 2
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Introduction - Motivation Ant Colony Optimization Motivation Natural Inspiration → based on the the behavior of ants seeking a path between their colony and a source of food Stigmergy Unorganized actions of individuals serve as a stimuli for other individuals by modifying their environment and result in a single outcome . In short: A group of individuals that behave as a sole entity. Katinka B¨ ohm 3
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Introduction - Motivation Ant Colony Optimization Motivation (contd.) I Swarm Intelligence method I probabilistic technique → non-deterministic I solve hard combinatorial optimization problems Definition Combinatorial Optimization Problem P = ( S , Ω , f ) S . . . finite set of decision variables, Ω . . . constraints, f . . . objective function to be minimized Prominent example: Traveling Salesman Katinka B¨ ohm 4
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Introduction - Metaheuristic Ant Colony Optimization Metaheuristic Ant Colony Optimization (ACO) Set parameters Initialize pheromone trails while termination condition not met do ConstructAntSolutions DaemonActions (optional) UpdatePheromones endwhile Katinka B¨ ohm 5
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Theoretical Approach - Ant System Ant Colony Optimization Ant System (AS) I oldest most basic algorithm I by Marco Dorigo in the 90s Ant Movement Probability for ant k to move from i to j in the next step: τ α ij · η β ij p k ij = il · η β P τ α ∀ c il il c il feasible where α and β control importance of pheromone τ vs. heuristic value η 1 Standard heuristic: η ij = d ij where d ij is the distance between i and j Katinka B¨ ohm 6
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Theoretical Approach - Ant System Ant Colony Optimization Ant System (AS) (cont.) Pheromone Update Pheromone update for all ants that have built a solution in that iteration: m X ∆ τ k τ ij ← (1 − ρ ) · τ ij + ij k =1 where ρ is the evaporation rate and ∆ τ k ij is the quantity of pheromone laid on edge ( ij ) with ij = Q ∆ τ k L k where Q is a constant and L k is the total length of the tour of ant k Katinka B¨ ohm 7
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Theoretical Approach - Max-Min Ant System Ant Colony Optimization Max-Min Ant System (MMAS) I pheromone values are bound I only the best ant updates its pheromone trails after solutions have been found Pheromone Update i τ max h (1 − ρ ) · τ ij + ∆ τ best τ ij ← ij τ min where ∆ τ best 1 = ij L best L best can be the iteration best or global best tour Katinka B¨ ohm 8
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Theoretical Approach - Ant Colony System Ant Colony Optimization Ant Colony System (ACS) I diversify the search through a local pheromone update I pseudorandom proportional rule for ant movement Local Pheromone Update Performed by all ants after each construction step to the last traversed edge τ ij = (1 − ψ ) · τ ij + ψ · τ 0 where ψ ∈ (0 , 1] is the pheromone decay coe ffi cient and ψ 0 is the initial pheromone value Pheromone Update ( (1 − ρ ) · τ ij + ρ · ∆ τ ij if ( i , j ) belongs to the best tour τ ij ← otherwise τ ij Katinka B¨ ohm 9
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Theoretical Approach - Overview Ant Colony Optimization Theoretical Approach Overview Algorithm Ant Movement Pheromones Update Evaporation τ ij ← (1 − ρ ) · τ ij + P m k =1 ∆ τ k Ant System (AS) random proportional all paths ij 1991 Max-Min Ant Sys- random proportional best-so-far tour tem (MMAX) min/max bound 2000 i τ max h (1 − ρ ) · τ ij + ∆ τ best τ ij ← ij τ min Ant Colony System pseudorandom local: τ ij = (1 − ψ ) · τ ij + ψ · τ 0 last step (ACS) proportional global: best-so-far tour 1997 ( (1 − ρ ) · τ ij + ρ · ∆ τ ij τ ij ← τ ij Katinka B¨ ohm 10
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Problem Types Ant Colony Optimization Problem Types I Routing Problems → Traveling Salesman, Vehicle Routing, Network Routing I Assignment Problems → Graph Coloring I Subset Problems → Set Covering, Knapsack Problem I Scheduling → Project Scheduling, Timetable Scheduling I Constraint Satisfaction Problems I Protein Folding Katinka B¨ ohm 11
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Robot Path Planning with ACO Ant Colony Optimization Robot Path Planning I NP -complete problem I static vs. dynamic environment I known vs. unknown environment I rerouting on collision I shortest path Katinka B¨ ohm 12
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Robot Path Planning with ACO Ant Colony Optimization Robot Path Planning Alg1 Mohamad Z. et al. [8] Shortest Path in a static environment Map Construction Generate a global free space map where the robot can traverse between the yellow nodes Free space nodes (white) can be traversed by the robot Katinka B¨ ohm 13
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Robot Path Planning with ACO Ant Colony Optimization Robot Path Planning Alg1 Mohamad Z. et al. [8] Ant Movement Probability p ij = η β ij · τ α ij with α = 5 , β = 5 1 Heuristic η = distance between next point with intersect point at reference line Katinka B¨ ohm 14
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Robot Path Planning with ACO Ant Colony Optimization Robot Path Planning Alg1 Mohamad Z. et al. [8] Pheromone Update local → after each step from one node to another global → after path calculation is finished Local Evaporation prevents accumulation of pheromone τ ij = (1 − ρ ) · τ ij with ρ = 0 . 5 Global Reinforcement (AS) τ ij = τ ij + P m k =1 ∆ τ k ∆ τ ij = Q ij , L k where Q . . . number of nodes L k . . . length of path chosen by ant k Katinka B¨ ohm 15
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Robot Path Planning with ACO Ant Colony Optimization Robot Path Planning Alg1 Mohamad Z. et al. [8] Results I comparison to a standard GA algorithm I ACO faster with smaller number of iterations (due to good state transition rule - distance to baseline) Katinka B¨ ohm 16
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Robot Path Planning with ACO Ant Colony Optimization Robot Path Planning Alg2 Michael Brand et al. [2] Shortest path in a dynamic environment I grid world of 20x20, 30x30 and 40x40 four possible movement directions: left, right, up, down I basic AS approach I re-routing after obstacles are added I focus on re-initialization of pheromones Global Initialization τ ij = 0 . 1 for every transition between blocks Local Initialization Gradient of pheromones around every object Pheromone levels are decreased in a cyclic fashion by a certain fraction (50%) Katinka B¨ ohm 17
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Robot Path Planning with ACO Ant Colony Optimization Robot Path Planning Alg2 Michael Brand et al. [2] Results Global Initialization Map Size 20x20 30x30 40x40 Iterations 151 277 148 Local Initialization: 1st iteration Path Length 39 66 138 Local Initialization Map Size 20x20 30x30 40x40 Iterations 122 84 69 Path Length 39 64 128 Local Initialization: 1000th iteration Katinka B¨ ohm 18
MIN-Fakult¨ at Fachbereich Informatik Universit¨ at Hamburg Analysis - Comparison Ant Colony Optimization Comparison to other meta-heuristic techniques I other techniques: Genetic Algorithms (GA), Simulated Annealing (SA), Particle Swarm Optimization (PSO), Tabu Search (TS) I hard to compare in general → dependent on specific problem instance, algorithm implementation and parameter settings ( No free lunch theorem ) I slow convergence compared to other approaches → long runtime for small easy instances and fast, pretty good results for complex instances I ACO often performs really bad or really good Katinka B¨ ohm 19
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