Part 1 Data Collection Planning

Transcription

1 N N Overview of the Lecture Data collection planning - (N), PC-(N), and (N)) Part Data Collection Planning Data Collection Planning al Problem Part I Traveling Saleman Problem () Jan Faigl Generalized Traveling Saleman Problem () Faculty of Electrical Engineering Czech Technical Univerity in Prague Lecture Orienteering Problem () BMUIR Artificial Intelligence in Robotic Jan Faigl, (N) BMUIR Lecture : Data Collection Planning N Part Data Collection Planning Traveling Saleman Problem with Neighborhood (N) Department of Computer Science / Jan Faigl, N Autonomou Data Collection BMUIR Lecture : Data Collection Planning N / Jan Faigl, N Prize-Collecting Traveling Saleman Problem with Neighborhood (PC-N) Having a et of enor (ampling tation), we aim to determine a cot-efficient path to retrieve data by autonomou underwater vehicle (AUV) from the individual enor (pli,pli+ ) T The data from i can be retrieved within δ ditance from i BMUIR Lecture : Data Collection Planning for δ = it i the PC- for ξ(i ) = and δ it i the N for ξ(i ) = and δ = it i the ordinary / Jan Faigl, N PC-N Example of Solution Solution cot PC-N Communication radiu ρ [km] / Jan Faigl, N Jan Faigl, Faigl and Hollinger (IROS, TNNLS ) BMUIR Lecture : Data Collection Planning / Jan Faigl, / N Σ = (σ,..., σn ), σi n, σi = σ for i = Branch and Bound, Integer Linear Programming (ILP) E.g., Concorde olver Approximation algorithm Minimum Spanning Tree (MST) heuritic with L Lopt Chritofide algorithm with L / L opt Heuritic algorithm Contructive heuritic Nearet Neighborhood (NN) algorithm -Opt local earch algorithm propoed by Croe 9 Lin-Kernighan (LK) heuritic E.g., Helgaun implementation of the LK heuritic Soft-Computing technique, e.g., Variable Neighborhood Search (VNS) Evolutionary approache Unupervied learning If c(i, ) = c(, i ) it i the Aymmetric The i known to be NP-hard unle P=NP Communication radiu ρ [km] N Exact olution Euclidean BMUIR Lecture : Data Collection Planning Exiting olver to the The can be conidered on a graph G (V, E ) where the et of vertice V repreent enor location S and E are edge connecting the node with the cot c(i, ) For implicity we can conider c(i, ) to be Euclidean ditance; otherwie, it i a olution of the path planning problem PC N SOM SOM + Computational time [m] N Let S be a et of n enor location S = {,..., n }, i R and c(i, ) i a cot of travel from i to Traveling Saleman Problem () i a problem to determine a cloed tour viiting each S uch that the total tour length i minimal, i.e., determine a equence of viit Σ = (σ,..., σn ) uch that! n minimize Σ L= c(σi, σi+ ) + c(σn, σ ) () i= PC N SOM SOM + BMUIR Lecture : Data Collection Planning Traveling Saleman Problem () Ocean Obervatorie Initiative (OOI) cenario S\ST PC-N include other variant of the Thee two apect (of general applicability) can be conidered in the Prize-Collecting Traveling Saleman Problem (PC-) and Orienteering Problem () and their extenion with neighborhood SOM PC-N N where ST S are enor uch that for each i ST there i pl on T = (pl,..., plk, plk ) and pl P for which (i, pl ) δ.. Data from the enor can be retrieved uing wirele communication / Find a cot efficient tour T viiting P uch that the total cot C (T ) of T i minimal C(T ) = c(pli, pli+ ) + ξ(), () Let the data collecting vehicle operate in R with the motion cot c(p, p ) for all pair of point p, p R Two practical apect of the data collection can be identified N The PC-N i a problem to Each enor ha aociated penalty ξ(i ) characterizing additional cot if the data are not retrieved from i. Data from particular enor may be of different importance PC-N Optimization Criterion Let n enor be located in R at the location S = {,..., n } E.g., Sampling tation on the ocean floor N Determine a et of unique location P = {p,..., pk }, k n, pi R, at which data reading are performed The planning problem i a variant of the Traveling Saleman Problem Jan Faigl, BMUIR Lecture : Data Collection Planning BMUIR Lecture : Data Collection Planning / Jan Faigl, BMUIR Lecture : Data Collection Planning /

2 MST-baed Approximation Algorithm to the Minimum Spanning Tree Heuritic. Compute the MST (denoted T ) of the input graph G. Contruct a graph H by doubling every edge of T. Shortcut repeated occurrence of a vertex in the tour For the triangle inequality, the length of uch a tour L i L L optimal, where L optimal i the cot of the optimal olution of the Chritofide Algorithm to the Chritofide algorithm. Compute the MST of the input graph G. Compute the minimal matching on the odd-degree vertice. Shortcut a traveral of the reulting Eulerian graph MST Matching Final tour For the triangle inequality, the length of uch a tour L i L L optimal, where L optimal i the cot of the optimal olution of the Length of the MST i L optimal Sum of length of the edge in the matching L optimal -Opt Heuritic. Ue a contruction heuritic to create an initial route NN algorithm, cheapet inertion, farther inertion. Repeat until no improvement i made. Determine wapping that can horten the tour (i, ) for i n and i + n route[] to route[i-] route[i] to route[] in revere order route[] to route[end] Determine length of the route Update the current route if length i horter than the exiting olution Jan Faigl, BMUIR Lecture : Data Collection Planning / Unupervied Learning baed Solution of the Senor location S = {,..., n}, R ; Neuron N = (ν,..., ν m), ν i R, m =.n Learning gain σ; epoch counter i; gain decreaing rate α =.; learning rate µ =.. N init ring of neuron a a mall ring around ome i S, e.g., a circle with radiu.. i ; σ.n +.;. I //clear inhibited neuron. foreach Π(S) (a permutation of S). ν argmin ν N \I (ν, ). foreach ν in d neighborhood of ν ν ν + µf (σ, d)( ν) { f (σ, d) = e d σ for d <.m, otherwie,. I I {ν } // inhibit the winner. σ ( α)σ; i i + ;. If (termination condition i not atified) Goto Step ; Otherwie retrieve olution preented location = (, ) i i, i, i, i, input layer connection weight output unit ν, ν, m m i+ Termination condition can be enor location i i = (, ) i, i, ( ν,, ν, ) i i+ ring of connected node Maximal number of learning epoch i i max, e.g., i max = Winner neuron are negligibly cloe to enor location, e.g., le than. Somhom, S., Modare, A., Enkawa, T. (999): Competition-baed neural network for the multiple travelling alemen problem with minmax obective. Computer & Operation Reearch. Faigl, J. et al. (): An application of the elf-organizing map in the non-euclidean Traveling Saleman Problem. Neurocomputing. Jan Faigl, BMUIR Lecture : Data Collection Planning / Traveling Saleman Problem with Neighborhood (N) Euclidean N with dik haped δ-neighborhood Sequence of viit to the region with particular location of the viit Jan Faigl, BMUIR Lecture : Data Collection Planning 9 / Jan Faigl, BMUIR Lecture : Data Collection Planning / Example of Unupervied Learning for the Learning epoch Learning epoch Learning epoch Learning epoch Jan Faigl, BMUIR Lecture : Data Collection Planning / Approache to the N A direct olution of the N approximation algorithm and heuritic E.g., uing evolutionary technique or unupervied learning Decoupled approach. Determine equence of viit Σ independently on the location P E.g., a the for centroid of the region R. For the equence Σ determine the location P to minimize the total tour length, e.g., Touring polygon problem (TPP) Sampling poible location and ue a forward earch for finding the bet location Continuou optimization uch a hill-climbing E.g., Local Iterative Optimization (LIO), Váňa & Faigl (IROS ) Sampling-baed approache For each region, ample poible location of viit into a dicrete et of location for each region The problem can be then formulated a the Generalized Traveling Saleman Problem () Euclidean N with, e.g., dik-haped δ neighborhood Simplified variant with region a dik with radiu δ remote ening with the δ communication range Jan Faigl, BMUIR Lecture : Data Collection Planning / Jan Faigl, BMUIR Lecture : Data Collection Planning / Traveling Saleman Problem with Neighborhood (N) Intead viiting a particular location S, R we can requet to viit, e.g., a region r R to ave travel cot, i.e., viit region R = {r,..., r n } The become the with Neighborhood (N) where it i neceary, in addition to the determination of the order of viit Σ, determine uitable location P = {p,..., p n }, p i r i, of viit to R The problem i a combination of combinatorial optimization to determine Σ with continuou optimization to determine P ( n ) minimize Σ,P L = c(p σi, p σi+ ) + c(p σn, p σ ) i= R = {r,..., r n}, r i R P = {p,..., p n}, p i r i Σ = (σ,..., σ n), σ i n, σ i σ for i Foreach r i R there i p i r i () In general, N i AP-hard, and cannot be approximated to within a factor ɛ, ɛ >, unle P=NP. Safra, S., Schwartz, O. () Jan Faigl, BMUIR Lecture : Data Collection Planning / Unupervied Learning for the N In the unupervied learning for the, we can ample uitable ening location during winner election We can ue the centroid of the region for the hortet path computation from ν to the region r preented to the network Then, an interection point of the path with the region can be ued a an alternate location For the Euclidean N with dik-haped δ neighborhood, we can compute the alternate location directly from the Euclidean ditance communication range δ δ p alternate location Faigl, J. et al. (): Viiting convex region in a polygonal map. Robotic and Autonomou Sytem. Jan Faigl, BMUIR Lecture : Data Collection Planning /

3 Example of Unupervied Learning for the N It alo provide olution for non-convex region, overlapping region, and coverage problem. Jan Faigl, BMUIR Lecture : Data Collection Planning / Sampling-baed Solution of the N For an unknown equence of the viit to the region, there are O(n k ) poible edge Finding the hortet path i NP-hard, we need to determine the equence of viit, which i the olution of the The decrite variant of the N can be formulated a the Jan Faigl, BMUIR Lecture : Data Collection Planning / Noon-Bean tranformation to tranfer to A Modify weight of the edge (arc) uch that the optimal A tour viit all vertice of the ame cluter before moving to the next cluter Adding a large a contant M to the weight of arc connecting the cluter, e.g., a um of the n heaviet edge Enure viiting all vertice of the cluter in precribed order, i.e., creating zero-length cycle within each cluter The tranformed A can be further tranformed to the For each vertex of the A created vertice in the, i.e., it increae the ize of the problem three time Noon, C.E., Bean, J.C. (99), An efficient tranformation of the generalized traveling aleman problem. INFOR: Information Sytem and Operational Reearch. q q R q q q R q R Solving the N a the TPP Iterative Refinement Let the equence of n polygon region be R = (r,..., r n ) Li, F., Klette, R.: Approximate algorithm for touring a equence of polygon.. Sampling the polygon into a dicrete et of point and determine all hortet path between each ampled point in the equence of the region viit E.g., uing viibility graph. Initialization: Contruct an initial touring polygon path uing a ampled point of each region Let the path be defined by P = (p, p,..., p n), where p i r i and L(P) be the length of the hortet path induced by P. Refinement: For i =,,..., n Find pi r i minimizing the length of the path d(p i, pi ) + d(pi, p i+), where d(p k, p l ) i the path length from p k to p l, p = p n, and p n+ = p If the total length of the current path over point pi i horter than over p i, replace the point p i by pi. Compute path length L new uing the refined point. Termination condition: If L new L < ɛ Stop the refinement. Otherwie L L new and go to Step. Final path contruction: ue the lat point and contruct the path uing the hortet path among obtacle between two conecutive point Jan Faigl, BMUIR Lecture : Data Collection Planning / Generalized Traveling Saleman Problem () For ampled neighborhood into dicrete et of location, we can formulate the problem a the Generalized Traveling Saleman Problem () Alo known a the Set or Covering Saleman Problem, etc. For a et of n et S = {S,..., S n}, each with particular et of location (node) S i = { i,..., i n i } The problem i to determine the hortet tour viiting each et S i, i.e., determining the order Σ of viit to S and a particular location i S i for each S i S ( n ) minimize Σ L = c( σi, σi+ ) + c( σn, σ ) i= Σ = (σ,..., σn), σi n, σi σ for i σi Sσi, Sσi = { σi,..., σi }, S nσ Sσi i In addition to exact, e.g., ILP-baed, olution, a heuritic algorithm GLNS i available (beide other heuritic) Smith, S. L., Imeon, F. (), GLNS: An effective large neighborhood earch heuritic for the Generalized Traveling Saleman Problem. Computer and Operation Reearch. Implementation in Julia Jan Faigl, BMUIR Lecture : Data Collection Planning / Example Noon-Bean tranformation (GA to A). Create a zero-length cycle in each et and et all other arc to (or M) To enure all vertice of the cluter are viited before leaving the cluter. For each edge (qi m, q n) create an edge (qm i, q n+ ) with a value increaed by ufficiently large M To enure viit of all vertice in a cluter before the next cluter q q R q q q R q R R q q q +M +M +M +M +M +M +M R q q +M q R Sampling-baed Decoupled Solution of the N Sample each neighborhood with, e.g., k = ample Determine equence of viit, e.g., by a olution of the E for the centroid of the region Finding the hortet tour take in a forward earch graph O(nk ) for nk edge in the equence Trying each of the k poible tarting location Jan Faigl, BMUIR Lecture : Data Collection Planning / Tranformation of the to the Aymmetric The Generalized can be tranformed into the Aymmetric that can be then olved, e.g., by LKH or exactly uing Concorde with further tranformation of the problem to the p, p, S p, p, p, p, S S p, p, S p, GA A tranformation of the to the A ha been propoed by Noon and Bean in 99, and it i called a the Noon-Bean Tranformation Noon, C.E., Bean, J.C. (99), An efficient tranformation of the generalized traveling aleman problem. INFOR: Information Sytem and Operational Reearch. Ben-Arieg, et al. (), Tranformation of generalized A into A. Operation Reearch Letter. Jan Faigl, BMUIR Lecture : Data Collection Planning / Example Noon-Bean tranformation (GA to A). Create a zero-length cycle in each et and et all other arc to (or M) To enure all vertice of the cluter are viited before leaving the cluter. For each edge (qi m, q n) create an edge (qm i, q n+ ) with a value increaed by ufficiently large M To enure viit of all vertice in a cluter before the next cluter q q R q q q R q R R q q q +M +M p, p, p, S S R q q +M q R Jan Faigl, BMUIR Lecture : Data Collection Planning / Jan Faigl, BMUIR Lecture : Data Collection Planning / Jan Faigl, BMUIR Lecture : Data Collection Planning /

4 Noon-Bean tranformation Matrix Notation. Create a zero-length cycle in each et; and. for each edge (qi m, q n ) create an edge (qi m, q n+ ) with a value increaed by ufficiently large M q q Original GA R q q q R q R q q q q q q q q q q q q repreent there are not edge inide the ame et; and denote unued edge q q q q q q q q q q q q Tranformed A q q q q q q q +M q +M q +M q +M q +M q +M +M +M Jan Faigl, BMUIR Lecture : Data Collection Planning / Orienteering Problem Specification Let the given et of n enor be located in R with the location S = {,..., n }, i R Each enor i ha an aociated core ζ i characterizing the reward if data from i are collected The vehicle i operating in R, and the travel cot i the Euclidean ditance Starting and final location are precribed We aim to determine a ubet of k location S k S that maximize the um of the collected reward while the travel cot to viit them i below T max The Orienteering Problem () combine two NP-hard problem: Knapack problem in determining the mot valuable location S k S Travel Saleman Problem () in determining the hortet tour Jan Faigl, BMUIR Lecture : Data Collection Planning / Benchmark Example of Solution Summary It tranform the GA into the A which can be further Solved by exiting olver, e.g., the Lin-Kernighan heuritic algorithm (LKH) the A can be further tranformed into the and olve it optimaly, e.g., by the Concorde olver It run in O(k n ) time and ue O(k n ) memory, where n i the number of et (region) each with up to k ample The tranformed A problem contain kn vertice Noon, C.E., Bean, J.C. (99), An efficient tranformation of the generalized traveling aleman problem. INFOR: Information Sytem and Operational Reearch. Jan Faigl, BMUIR Lecture : Data Collection Planning / Orienteering Problem Optimization Criterion Let Σ = (σ,..., σ k ) be a permutation of k enor label, σ i n and σ i σ for i Σ define a tour T = ( σ,..., σk ) viiting the elected enor S k Let the tart and end point of the tour be σ = and σ k = n The Orienteering problem () i to determine the number of enor k, the ubet of enor S k, and their equence Σ uch that maximize k,sk,σ R = i= ζ σi ( σi, σi ) T max and i= σ =, σk = n. The combine the problem of determining the mot valuable location S k with finding the hortet tour T viiting the location S k. It i NP-hard, ince for = n and particular S k it become the. Jan Faigl, BMUIR Lecture : Data Collection Planning / Unupervied Learning for the / () The Orienteering Problem () The problem i to collect a many reward a poible within the given travel budget (T max ), which i epecially uitable for robotic vehicle uch a multi-rotor Unmanned Aerial Vehicle (UAV) The tarting and termination location are precribed and can be different Travel budget T max =, Collected reward R = 9 The olution may not be a cloed tour a in the Travel budget T max =, Collected reward R = Jan Faigl, BMUIR Lecture : Data Collection Planning / Exiting Heuritic Approache for the The Orienteering Problem ha been addreed by everal approache, e.g., RB -phae heuritic algorithm propoed in [] PL Reult for the method propoed by Pillai in [] CGW Heuritic algorithm propoed in [] GLS Guided local earch algorithm propoed in [] [] I.-M. Chao, B. L. Golden, and E. A. Wail. A fat and effective heuritic for the orienteering problem. European Journal of Operational Reearch, (): 9, 99. [] R. S. Pillai. The traveling aleman ubet-tour problem with one additional contraint (TSSP+ ). Ph.D. thei, The Univerity of Tenneee, Knoxville, TN, 99. [] R. Rameh and K. M. Brown. An efficient four-phae heuritic for the generalized orienteering problem. Computer & Operation Reearch, ():, 99. [] P. Vanteenwegen, W. Souffriau, G. V. Berghe, and D. V. Oudheuden. A guided local earch metaheuritic for the team orienteering problem. European Journal of Operational Reearch, 9():, 9. Jan Faigl, BMUIR Lecture : Data Collection Planning 9 / Unupervied Learning for the / Tmax =, R= Tmax =, R = Tmax =, R= A olution of the i imilar to the olution of the PC- and We need to atify the limited travel budget T max, which need the final tour over the ening location During the unupervied learning, the winner are aociated with the particular ening location, which can be utilized to determine the tour a a olution of the repreented by the network: The winner election for S i conditioned according to T max The network i adapted only if the tour T win repreented by the current winner would be horter or equal than T max L(T win ) ( νp, νn ) + ( νp, ) + (, νn ) T max The unupervied learning perform a tochatic earch teered by the reward and the length of the tour to be below T max Tmax =9, R=9 Tmax =9, R= Tmax =, R= Learning epoch Learning epoch Learning epoch Final olution Thi i utilized in the conditional adaptation of the network toward the ening location and the adaptation i performed only if the tour repreented by the network after the adaptation would atify T max Epoch, R= Epoch, R= Epoch, R= Final olution, R=9 Jan Faigl, BMUIR Lecture : Data Collection Planning / Jan Faigl, BMUIR Lecture : Data Collection Planning / Jan Faigl, BMUIR Lecture : Data Collection Planning /

5 Comparion with Exiting Algorithm for the Standard benchmark problem for the Orienteering Problem repreent variou cenario with everal value of T max The reult (reward) found by different approache preented a the average ratio (and tandard deviation) to the bet-known olution Intance of the Tiligiride problem Problem Set RB PL CGW Unupervied Learning Set, T max.99/../../../. Set, T max./..99/..99/..99/. Set, T max./../../../. Diamond-haped (Set ) and Square-haped (Set ) tet problem Problem Set RB PL CGW Unupervied Learning Set, T max.9/../..99/..9/. Set, T max.9/../..99/..9/. Required computational time i up to unit of econd, but for mall problem ten or hundred of milliecond. Jan Faigl, BMUIR Lecture : Data Collection Planning / Generalization of the Unupervied Learning to the The ame idea of the alternate location a in the N Similarly to the with Neighborhood and PC-N we can formulate the. T max =, δ=., R= T max =, δ=., R= Jan Faigl, BMUIR Lecture : Data Collection Planning / Influence of the δ-sening Ditance Influence of increaing communication range to the um of the collected reward Data collection uing wirele data tranfer allow to reliably retrieve data within ome communication radiu δ Dik-haped δ-neighborhood We need to determine the mot uitable location P k uch that maximize k,pk,σ R = i= ζ σi (p σi, p σi ) T max, i= (p σi, σi ) δ, p σi R, p σ =, p σk = n. T max =, R = Introduced by Bet, Faigl, Fitch (IROS, SMC, IJCNN ) More reward can be collected than for the formulation with the ame travel budget T max Jan Faigl, BMUIR Lecture : Data Collection Planning / with Neighborhood (N) Example of Solution Diamond-haped problem Set SOM olution for T max and δ p p communication range δ δ p alternate location The location p for retrieving data from i determined a the alternate goal location during the conditioned winner election Jan Faigl, BMUIR Lecture : Data Collection Planning / Summary of the Lecture Jan Faigl, BMUIR Lecture : Data Collection Planning / Problem Solution of the R bet R SOM Set, T max = Set, T max = Set, T max = 9 Allowing to data reading within the communication range δ may ignificantly increae the collected reward, while keeping the budget under T max Collected reward - R Tiligiride Set, T max= Diamond haped Set, T max= Square haped Set, T max= Communication range - δ Jan Faigl, BMUIR Lecture : Data Collection Planning / Data Collection Planning motivational problem and olution Prize-Collecting Traveling Saleman Problem with Neighborhood (PC-N) Traveling Saleman Problem () Approximation and heuritic approache Traveling Saleman Problem with Neighborhood (N) Sampling-baed and decoupled approache Unupervied learning Generalized Traveling Saleman Problem () Heuritic and tranformation ( A) approache Orienteering problem () Heuritic and unupervied learning baed approache Orienteering problem with Neighborhood (N) Unupervied learning baed approach Next: Data-collection planning with curvature-contrained vehicle Jan Faigl, BMUIR Lecture : Data Collection Planning / T max =, δ=., R= T max =, δ=., R= T max =, δ=., R= Square-haped problem Set SOM olution for T max and δ T max =9, δ=., R= T max =, δ=., R= T max =, δ=., R= In addition to unupervied learning, Variable Neighborhood Search (VNS) for the ha been generalized to the N Jan Faigl, BMUIR Lecture : Data Collection Planning 9 /